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Submitted on 29 Apr 2014

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Structures de Poisson Logarithmiques : invariantscohomologiques et préquantification

Joseph Dongho

To cite this version:Joseph Dongho. Structures de Poisson Logarithmiques : invariants cohomologiques et préquantifica-tion. Analyse classique [math.CA]. Université d’Angers, 2012. Français. tel-00985181

♥♥é

♦rr

❯❱

♦ ♦t♦r ❯

ès ♦t♦rt ❯♥rsté ♥rs

♥t♦♥ té♠tqs

Prés♥té ♣r

♦s♣ ♦♥♦

♥r à

à ❯♥rsté ♥rs à ♥rs

trtrs P♦ss♦♥ ♦rt♠qs ♥r♥ts♦♦♠♦♦qs t ♣réq♥tt♦♥

r②

♣♣♦rtrs

r♠♥♦ r Pr♦ssr ♥ss r③ rr♦ Pr♦ssr ❯

①♠♥trs

♥ ♦♠s Pr♦ssr ♥rs♥ r♥r Pr♦ssr ♥rs

è ♦ ♦②♦♠ Pr♦ssr ♦♥t♣r

rtr ès ❱♦♦② ♦ts♦ Pr♦ssr ♥rs♦♥rr t♦♥ ♦♠♦ Pr♦ssr ❯♥ ❨♦♥é

é♣rt♠♥t té♠tqs ❯♥rsté ♥rs ❯

♦r ♦sr ①

➒ ♥♦tê ♠ á♠

♠r♠♥ts

r♠r s Pr♦ssrs ♥ ♦♠s t t♦♥ ♦♠♦ q

♦♥t éé ♠♦♥ ♥térêt ♣♦r ♦♠♥ t♦♣♦♦ érq t ♠♦♥t t

♦♥♥îtr ♦rt♦r ♥♥ r ♥ té♠tqs ♦ù

tt tès été té é♠r à é♠r

♦t ♠ rtt Pr♦ssr ❱♦♦② ♦ts♦ q ♠

♦rt♦r t ♣té ♥rr ♠s tr① P♥♥t t♦t tt ♣ér♦

é♥éé s ♦♥t♥ tt♥t♦♥ qté ss ①♣t♦♥s t ss

ss ♦♥ss sés s rr s ♣r♦è♠s réés ♣r st♥ t ♦ssé

♥♠érq q ♥♦s sé♣rt q♥ éts ♠r♦♥ ♥éstt ♣s à ♠

♦♥r ♣r téé♣♦♥ q♥ ss ♠s rst♥t s♥s ré♣♦♥s ê♠ ss ♠s

t ss ♥s ♥s s s ❯r♥♥s t ss ♥ ♠♣ê♥t ♣s

r ♠s ♣r♦ts ♠ s♦♥s q ♣r ♣s ♥ ♦s t ♦é r ♥

❯r♥ à ♦s♦ ♣♦r ♠ ♦♥ttr t ré♣♦♥r à ♠s qst♦♥s

été étr♠♥♥t ♥s tr ♥ tr♦ ♥ t qt ♥ r♥çs

♣♦♥t ♠ ♣r♠ttr ①♣r♠r q rss♥t ♣r♦♦♥ ♠♦ ♥ s♦♥ st

♦t t ♣r ♠r à r♥r ♥t♣ ss q

r t ♠① q q ♦♥q r ♣r q st q r♣rés♥t

♣♦r ♠♦ ♥ r♠r s♥èr♠♥t

P♦r r r♠r é♠♥t t♦s s ♠♠rs ♦rt♦r

q ôt♦②és q♦t♥♥♠♥t r à ♦r ❳ ♠s

❨ ♦♥ ❯ r♥ç♦s ❱ r♥t

❯ r♥r réér ♥P♣♣ ♥

❯ ♥ ❱ s ❩❨ P♦tr ❯ ♦ ❯

♦ï ❲ ♦♣ ♦♦♣ ♥qs

❯❯ ést♥ P❯ s♥ ❱❱

r ❯ P♣♣ ♦

♠♠ ❨ è ❨❱ éè♥ q ♦♥t

♦♥♥é ♣♦t♥t à ♠ tr ♠té♠tq à trrs rs rs ①♣♦sés

r♠r ♣rtèr♠♥t s Pr♦ssrs ♥qs ♦r

t♦r t ❯ ♦ï rtr ♦♥t ♣♦r t♦ts s

tés ♠♥strts t r ♥ ♥s trt♠♥t ♠s ♦ssrs rs

srs étr♥♠♥t r♦♥♥ss♥t

r♠r

s srétrs ♦rt♦r t ♥♦s ♦térs s ❯

tr♥ ❯❩ ①♥r Ptt♦r♣s r♥ç♦s r♦♥

♠

s rtrs é♣rt♠♥t P♣♣ ❯ ♥ ❯ t

r♥ç♦s ❯

s rtrs ♦rt♦r ♥ ♠ P❯

t ♥q

♥é♥r s②stè♠ q♥

r♠r ♣rtèr♠♥t ♥ q ♣♣r♦ ♥ é♠

r ♠♦♥ ♥sr♣t♦♥ ♥ tès ♦rt♦r t q ♠ré ss ♦♣t♦♥s

tr♦t t♦♦rs ♥ t♠♣s ♣♦r ♣♣♦rtr s ①♣t♦♥s sr s ♣♦♥ts ♦♠r

♣r♦① rt t♦ st rsté r♥t t♦t t♠♣s tt♥t à ♠s

qst♦♥s s ♣r♥t ♣r♦s ss ♦t♦♥s q♦t♥♥s ♣♦r tr♦r s

ré♣♦♥ss à ♠s ♥s ♥♦rs ♠s s♦♥ ♠ érr ♥s

q ♠ tr♥s♠t ❱♦♦② s s♦t♦♥s à ♠s ♣r♦è♠s ♥ ♥r

sqs t ♦t♥ ♣rès ♥ ss ♦♦rtrs ♥ s♣♥ ♦♥ r

♠étt t♦♦rs ♦rt s♦t ♦♦r ♦ù à s à é ♠é♦tt t

♠♣♣♦rtt s ①♣t♦♥s ♥éssrs ♠rr ♠ ♠♥r s ♥st ♣s

♠♦♥ tr rtr tès ss ♦ r♠ttr ♥t♣

r♠r ♥♥ ♦r ♣té êtr ♠♠r r② tt tès

r♠r é♠♥t Pr♦ssr è ♦ ♦②♦♠ q ♣t ♠♦♥

tt♥t♦♥ ♥rs é♦♠étr à trrs s♦♥ ♥♦ ①♣♦sé sr ❱èr à

①è♠ r♥♦♥tr r♦♣ ♦♣♦♦ t é♦♠étr rq ♥tr à

❯♥rsté s♥ ♥ é♠r ê♠ ♠ t t♦s ♠s ♣r♦ts

t ♠ tr♥s♠s ♥ ♣rès ♥ s ♦rrt♦♥s ♥éssrs r♠r ♣♦r

s♦♥ ♥tt♦♥ t q ♠ résré ♦♠r à ❯

❯ P ♥ ♣r♦t

♣♦r r♠rr t♦s s ♠♠rs ♦rt♦r ♣♦r r r♥t

♦rt sé♦r r♠r ♥♥ ♦r ♣té r ♣rt s ♠♠rs r②

r♠r Pr♦ssr r♠♥♦ r ♣♦r ♥térêt q ♣♦rté à s

tr① t ♣♦r ♦r ♣té tâ r♣♣♦rtr ss ♣rtèr♠♥t

r♦♥♥ss♥t ♣♦r ss ①♣t♦♥s st s s♦t♦♥s ♣tqs s éqt♦♥s

P râ à r♦s ♦r ♦♠♣rs s ♥♦t♦♥s rétés rr t

rêt♠♥t t♥♥t ♥♦rs ♠s ♥♦s sss♦♥ ss ♦

s r♥r ♥t♣

r③♦ Pr♦s♦r s r③ r♦ ♣♦r ss tr♦s ♠út♣s s♦r ♦s

s♦rs rs ② ♣r r ♣t♦ ♣sr ♦s ♣③♦s r♦s trr st

tss s rs sñ♦r Pr♦s♦r

r♠r é♠♥t ♦tr ♥ P♣ r ♦♠é♦ ♣♦r s♦♥ à

♥rs ♥ ss ♠t♣s ♦♣s s t ♠s s ♥♦r♠♥ts ♠t♣s

♠♦♥t été ♥ ♠♣♦rt♥ ♣t ♥s tr

r♠r ♠s ♠rs ♦t♦r♥ts ①♥r ❯

♠r ❯ ③③ ❯ ♦♠♥ é♠ é♠ ♥

①② r ♦♣ ❨ ♥sts ② ❯ ♦♠s③

r♥ P ♥r② P❯ ❯ ❨ ❲ ❳

♦♥ r ❨ s ♦♥t s ♠♦♠♣♥é t♦t ♦♥

s r♥èrs ♥♥és r♠r ♣rtèr♠♥t ①♥r

P❯ t ③♥♥ st♦♥ ♣♦r ♠♦r é à ♦rrr t①t r♠r

Pstr st♦♥ ♦r② ♥♥tt sr ♦ t t♦s s rèrs s

♥éq ♣♦r r s♦t♥ s♥s t rs ♠t♣s ♣rèrs r♠r

ss ♠s ♥♥s ♦ès ②é ❨♦♦ ♣♦r r s♦t♥ t ♥♦r♠♥ts

♣♥s ♣rtèr♠♥t à ♦ ♦rt♥ ♦♠s ♦♥ ♠

r r♠r ♣r♦♦♥é♠♥t rèr ♦♥♦ ♦s♣ t ♦r tt ♣♦r

r ♣rèrs t é♥ét♦♥s

s tr① ♥r♥t ♠s ♦ts s♥s ❯ ss

♠♥ ♠ ♣r♦♥♥r ♣♦r t♦s ♠s ér♠♥ts s ♦♦♥té s♦♠♣ss

à t♦s s âs t ♥ t♦t t♠♣s

r♠r s ♦r♥strs ♣r♦t ♣♦r r s♦t♥ s♥s

t♥s à r♠rr ♣rtèr♠♥t r ♥P♦rt ♥ès ♦♠③

♥♥ t ② ❱♥♥t ♣♦r rs ♠t♣s ♥tr♥t♦♥s r♥t s tr①

r♠r t♦s s ♣rs♦♥♥s ♠ss r♥ ♠r♦♥ ♣♦r s

♠t♣s s ♦rés

t♥s é♠♥t à ①♣r♠r ♠s r♠r♠♥ts ♥rs s trs ♠r♦

❯ ❯♥rsté ❨♦♥é r ♥ ♦ ❯♥rsté

r♦ ♣♦r rs s♦t♥s ♦♥t♥s r♠r Pr♦ssrs ♦ ss

rtr ♦ ♦r♠ ♣érr r♦ ♦tr ♦r♠♥

é♣rt♠♥t té♠tqs r♦ Pr♦ssr

r♥ç♦s ❲♠♦♥ ♦ ♦rrt ♦t♠ é♣rt♠♥t

té♠tqs ❯♥rsté ❨♦♥é ♦tr ♠ t t♦s

s ♠♠rs é♣rt♠♥t té♠tqs ❯♥rsté ❨♦♥é

r♠r ♠s ♦ès ①s rt ♦r r r♥♥ Prr ♦ t

t♦s s ♣rs♦♥♥s r♦

r ss à ♥♥ ♦ ♦♥ ♠ t♥r é♣♦s ♣♦r s ♥♥és

sr tt tès ♥♥ ♦♠é ssé ♣r ss r♠s ♥♦tr♥s

r♠r ss ♠♦♥ ♦♥ ③♠ ♥r t s♦♥ é♣♦s ♣♦r r ♥tt♦♥

à ♦s s♦r r♠r ♥♥ ♠♥ ♥♥ ♠♠♦ t P♣ ♦s♣

♠♦ ♣♦r r ♣rèrs ♦♥t♥s ♣♥s s♥s ss à ♠ é♥t r♥ ♠èr

q ♠♥ s♥s ss à ♠ ♦rr ♦♥r t q ♠♣♣rs ♣t♥

t ♣r♦♥

r♠r ♣rtèr♠♥t P♣ ♥ st♥ ♣♦r t♦t s♦♥ s♦t♥ t ss

♥♦♠r① ♥♦r♠♥ts ♦♠♠ttrs ♥ ♣éé ♠♣r♦♥♥ s ♦s

♠♠♥ tt t P♣ ♦♠ss ts ss q ss ♣ssé ♥♣rç

s r♥èrs ♥♥és s ♦♠♣r♥r♦♥t ♥♠♥t ♣♦rq♦ ♥s ①

♥ srs r♥ ♥♦rs ♣s à ♦r♥♥ t ♠♠♥ ③♥♥ r

♥ ♣♥sé ♣s ♣♦r r♣♦s q r♠♣ç ♠ é♥t r♥ ♠èr

♠s q ♠rs♠♥t t é ♣r ♥ ♦tr ♥s s ♣r♦♣r s♥

t♦ ♠♠♥ ♠♥ q trr ♥♦s ♥êtrs t s♦t éèr

r ♥ s♦♥r ♥♦ ♠ q s♥s ss ♠♥♦r

t t ♠ ♣r♠ttt rtr♦r t♦♥ ♠ t ♠tr♥ q ♠

♠♥qt ♥ s t♠♣s ♥ ♣ssés à ♥rs s ♣♦r ♠♦ ♣s q♥ ♠èr

t r♠r ♠♦r ♦♥♥é ♥ ♣ ♥s t♦♥ ♦②r ♠ ♦♥♥r s♥s

♦t ♥ ♠♦②♥♥ étr♥sr ts t♦♥s ♥rs ♠♦ ss ♦ très t

é♥r t♦s ① q ♦♥t é t♦qé à ♦tr ♣♦rt râ à ♦s ♠s sé♦rs à ♥rs

♦♥t été ♠ ♠♥s♠ t t♥rss r♠r éé♥ t t ❱♦♦②

♠♦r ♣té ♦♠♠ rèr t ♥♦♥ ♦♠♠ étr♥r ♦s s♦t ♥ ♥r

♠rs t ♦♠é râs ss q r r ♥ ♣r♦r ♠r♦♥s

t ♦♥ ♥ ♣t r♠ssr s ♠♥s ♣ ♥ ♣r♥r s♦é

é♦t s ♣rèr s s ① ♠ t ♥ t ♦r à tt

♠ t♦ts s ♦♥♥s râs q ♠ért ♣r♥ s étrsss ♥ ♠♦♥

♥♥♠é ♣r s♣tr ♠ t sr s s♥trs t♠t① ♥ q

♦ss ♦♠♠ t s ♣r♦♠s à ♥♦s ï① s é♥ét♦♥s ♠♣ts à r rté

r♠r ♥♥ ♠s ♥♥ts ♠♦ ❱rs ♠ts ③♠

r s r♦ ♠ ♦ts♦ t ♦t r ♣♦r rs

♣rèrs s♥s ss à ♠♦♥ ér ♥♦rs ♣s ♠s s♦rs t rèrs ♠♦

r♥st♥ ♦ ttr♥ ♦t ②sé r♥r ♦♥♥

Ps ♠♥ ♦♥t♥ t♥♦ ♠ ♦tt ♥♠ r♥ ♥t

ttr♥ ♣♦r r ♥♦♠r① srs à ♠♦♥ ♥r♦t r ♥ très r♥

♠rt♦♥ à ér ♠s t♥ts à ♦♥ à ③ à ♦

r♠r ♠♦♥ ♦s♥ Pr♦ssr ♠♦ rt Ps t t♦t s ♠ ♣♦r

r s♦t♥ s♥s

s ♠tèrs

♥tr♦t♦♥ é♥ér

st ès

t ès

ts t ♠ét♦s tr

♦♥strt♦♥ ♠♦ s ér♥ts ♦r♠s ♦rt

♠qs

trtrs P♦ss♦♥ ♦rt♠qs t ♦♦♠♦♦ P♦s

s♦♥ ♦rt♠q

Préq♥tt♦♥ ♦rt♠q

s réstts ♣r♥♣①

♦♥strt♦♥ s èrs P♦ss♦♥ ♦rt♠qs

♦♥strt♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q t

qqs ①♠♣s

♣♣t♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦tés

r s ♦r♠s ér♥ts ♦rt♠qs

r s strtrs P♦ss♦♥ ♦rt♠qs t ♥r♥ts ♦

♦♠♦♦qs

Prs♣ts

r s èrs P♦ss♦♥ ♦rt♠qs

r s ♦r♠s ér♥ts ♦rt♠qs

r q♥tt♦♥

r s strtrs P♦ss♦♥ ♦rt♠qs

èrs P♦ss♦♥ ♦rt♠qs

ért♦♥s ♦rt♠qs

ér♥ts ♦r♠s ♦rt♠qs

éré ♣r r♣♣♦rt à ♥ ért♦♥ ♦rt♠q

trtrs èrs P♦ss♦♥ ♦rt♠qs

qs ①♠♣s èrs P♦ss♦♥ ♦rt♠qs

❱rétés P♦ss♦♥ ♦rt♠qs

sr r

❯♥ r♠rq sr s ♦r♠s ér♥ts ♦rt♠qs t

♠♣s trs ♦rt♠qs

é♥t♦♥ t ♣r♠èrs ♣r♦♣rétés

❱rétés ♦s②♠♣tqs

qs ①♠♣s rétés P♦ss♦♥ ♦rt♠qs

s♣ s SU(2) ♠♦♥♦♣ôs ♠♥étqs r

s ♠tèrs

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

èrs ♥rt ♦rt♠qs

trtr èr ♥rt sr ΩA(log I) ♥t ♣r♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I.

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

qs strtrs èr ss♦és ① strtrs

P♦ss♦♥ ♦rt♠qs

trtrs èr ♥rt sr ΩX(logD)

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠qs s str

trs ♦s②♠♣tqs

♦♦♠♦♦ P♦ss♦♥ t P♦ss♦♥ ♦

rt♠q strtr P♦ss♦♥ x, y = 0, x, z =

0, y, z = xyz sr A = C[x, y, z]

Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

qs ♣r♦♣rétés s strtrs ♦s②♠♣tqs

♦♥♥①♦♥ ♦rt♠q

♥trté s ♦r♠s ♦rt♠qs r♠és

Préq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt♠qs

qs r♠rqs sr ♦♦♠♦♦ s rétés P♦ss♦♥

♦rt♠qs

ss r♥P♦ss♦♥ ♦rt♠q

①♠♣s ♣♣t♦♥s

Préq♥tt♦♥ (C2, π = z1∂z1 ∧ ∂z2)

Préqtt♦♥ CP1 ♠♥ strtr

P♦♥ts ét qqs é♠♦♥strt♦♥s

s ♣♦♥ts s qqs s

s strtr f, g = xyzdf ∧ dg ∧ dpdx ∧ dy ∧ dz

♦♥tr♦♥s q ∂1 ∂0 = 0

♦♥tr♦♥s q ∂2 ∂1 = 0

s strtr P♦ss♦♥ x, y = x.

♦r♣

♣tr

♥tr♦t♦♥ é♥ér

♦♠♠r st ès

t ès

ts t ♠ét♦s tr

♦♥strt♦♥ ♠♦ s ér♥ts ♦r♠s ♦rt

♠qs

trtrs P♦ss♦♥ ♦rt♠qs t ♦♦♠♦♦ P♦s

s♦♥ ♦rt♠q

Préq♥tt♦♥ ♦rt♠q

s réstts ♣r♥♣①

♦♥strt♦♥ s èrs P♦ss♦♥ ♦rt♠qs

♦♥strt♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q t

qqs ①♠♣s

♣♣t♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦tés

r s ♦r♠s ér♥ts ♦rt♠qs

r s strtrs P♦ss♦♥ ♦rt♠qs t ♥r♥ts ♦

♦♠♦♦qs

Prs♣ts

r s èrs P♦ss♦♥ ♦rt♠qs

r s ♦r♠s ér♥ts ♦rt♠qs

r q♥tt♦♥

st ès

♦t X ♥ rété ♦♠♣① ♠♥s♦♥ ♥ n t D ♥ sr rét

X éqt♦♥ h = 0 ♦ù h st r♠ ♥ ♦♥t♦♥ ♦♦♠♦r♣ ♥ ♥♦t OX

s s r♠s ♦♥t♦♥s ♦♦♠♦r♣s sr X. ❯♥ strtr P♦ss♦♥

♦♦♠♦r♣ sr X st ♦♥♥é ♥ r♦t −,− q ss♥ à ♥ ♦♣ (f, g)

r♠s ♦♥t♦♥s ♦♦♠♦r♣s ♥ ♥ ♣♦♥t x X ♥ r♠ f, g ♦♥t♦♥

♦♦♠♦r♣ ♥ x ér♥t s ♣r♦♣rétés s♥ts

• −,− st ♥ér ♥ts②♠étrq

• f, g, h+ g, h, f+ h, f, g = 0 ♥tté ♦

♣tr ♥tr♦t♦♥ é♥ér

• f, gh = f, gh+ f, hg rè ♥③

r♥t ♠ê♠ é♥r ♥ ♠♣ trs q ♦♥ ♣t érr ♥s ♥

s②stè♠ ♦♦r♦♥♥és ♦s

P =1

2

∑1≤i,j≤n

Pij(x)∂

∂xi∧ ∂

∂xj

=∑

1≤i<j≤nPij(x)

∂

∂xi∧ ∂

∂xj Pij = −Pji

t q ér ♥tté ♦

∑

1≤i<j≤n

(Pil∂Pjk∂xl

+ Pjl∂Pki∂xl

+ Pkl∂Pij∂xl

) = 0

♣♦r 1 ≤ i, j, k ≤ n. ♥ é♥t ♦rs r♦t P♦ss♦♥ ♦♦♠♦r♣ ♣r

f, g := 〈P, df ∧ dg〉 =∑

1≤i<j≤n

Pij(x)(∂f

∂xi

∂g

∂xj− ∂g

∂xi

∂f

∂xj).

ts strtrs ♥s♥t ♦r ❬P♦s ❪ ♥ ♦♠♦♠♦r♣s♠ OX

♥ér H : ΩX → DerX(OX) t q H(df)(g) = f, g. H st ♣♣é ♣♣

t♦♥ ♠t♦♥♥♥ ss♦é à P. ♠t♦♥♥ ss♦é à t♦t r♠ ♦♥t♦♥

♦♦♠♦r♣ f rt♠♥t à P st r♠ ♠♣ tr é♥ ♣r

Xf = H(df) =n∑

i=1

xi, f∂

∂xi.

Pr rs ♥ r♠ ♠♣ trs δ st t ♦rt♠q ♦♥ D

♦r ❬t♦ ❪ s δ(h) ∈ hOX . ♥ ♥♦t DerX(logD) s r♠s

♠♣ trs ♦rt♠qs ♦♥ D. ♥ ♠♦♥tr q DerX(logD) st

st ♣♦r r♦t ♠♣ trs

❯♥ strtr P♦ss♦♥ ♦♦♠♦r♣ P sr X sr t ♦rt♠q ♦♥ D

s ♠t♦♥♥ ss♦é à t♦t r♠ ♦♥t♦♥ ♦♦♠♦r♣ f st ♥ st♦♥

DerX(logD). ts strtrs P♦ss♦♥ s♦♥t ♥ é♥érst♦♥ s strtrs

P♦ss♦♥ ♥ts ♣r s strtrs ♦s②♠♣tqs ♦s r♣♣♦♥s q s ♥

♦tr ♠♥s♦♥ X st ♣r ♥ ♦r♠ ♠ér♦♠♦r♣ ω st t ♦s②♠♣

tq sr X s st ♦rt♠q r♠é t ♥♦♥ éé♥éré ♥s s ω st ♥

♦r♠ ♦s②♠♣tq sr X, ♦rs ♣♦r t♦t r♠ ♦♥t♦♥ ♦♦♠♦r♣ f, g,

r♦t

f, g = ω(Xf , Xg)

♦ù iXfω = −df st P♦ss♦♥ ♥ ♣♣ r♦t P♦ss♦♥ ♦s②♠♣tq

s strtrs ♦s②♠♣tqs s♦♥t tsés à s ♥ rss ♥s s réér♥s

❬r ❱rr ❪ ❬♦t♦ ❪ t ❬t♦ ❪

t ès

t ès

♦s ♦ts s♦♥t

• ♥tr♦r s ♥♦t♦♥s èr P♦ss♦♥ t rété P♦ss♦♥ ♦rt♠q

• r♠♣r ♥s ♣r♦sss ♣réq♥t♦♥ s♣ s ♣ss ssqs

♣r ♥ rété P♦ss♦♥ ♦rt♠q

• ♥tr♦r s ♦♦♠♦♦s P♦ss♦♥ ♦rt♠q t s♥ srr ♣♦r

étr ♣réq♥tt♦♥ t②♣ rété

♣♦♥t ♠té♠tq ♣réq♥tr ♥ rté s②♠♣tq (X,ω) st

étr ♥ ♦rrs♣♦♥♥ ϕ ♥tr èr (F(X) ⊂ C∞(X), −,−) s

♦srs ssqs t ♥ s♣ rt H à ♦♥strr ♦ù −,− és♥

strtr P♦ss♦♥ ♥t ♣r ω ♣rès r tt ♦rrs♣♦♥♥ ♦t

stsr s ♣r♦♣rétés s♥ts

ϕ st t

s f st ♥ ♦sr ♦♥st♥t ♦rs ϕ(f) st ♠t♣t♦♥ ♣r f.

[f1, f2] = f3 ♦rs ϕ(f1)ϕ(f2) − ϕ(f2)ϕ(f1) = −ihϕ(f3) ♦ù h és♥ ♦♥

st♥t P♥

tr♠♥t t ϕ ♦t r♥r ♦♠♠tt r♠♠ èrs ♥rt

s♥t

0 // F(X)m // +

1 (Γ(L))σ // DerX // 0

0 // R //

OO

(F(X), ω)

ϕ

OO

// Ham(F(X))

OO

// 0.

♦♥ ♦r ❬❯r♥ ❪

ϕ(as) = ∇v(a)s+ 2iπas

♦ù ∇ és♥ ♦♥♥①♦♥ sr ♥ ré ♥ r♦t ♦♠♣① p : L→ X t +1 (Γ(L))

♠♦ s ♦♣értrs ér♥ts ♦rr ♥érr ♦ é à 1 sr ♠♦ s

st♦♥s L.

ts t ♠ét♦s tr

②♥t ♠♦é ♥tr s s♣s ♣ss ♦♥♥t ♣♣♦rtr s ♠♦

t♦♥s s♦t sr s t♥qs ss s♦t s ♦♥srr t ♠♦r s ♦ts

♦s ♦♣t♦♥s ♣♦r r♥èr ♠ét♦ P♦r ♥♦s ♥tr♦s♦♥s ♥♦t♦♥

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q râ à q ♥♦s ♠sr♦♥s ♦strt♦♥

à ①st♥ H

♦s r♣rtss♦♥s tr ♦♠♠ st

♣tr ♥tr♦t♦♥ é♥ér

♦♥strt♦♥ ♠♦ s ér♥ts ♦r♠s ♦rt♠qs

Prt♥t ♥ é ♣r♦♣r I ♥♥ré ♣r ♥ ♣rt S = u1, ..., up ♥

èr ♦♠♠tt t ♥tr A ♥té 1A ♥♦s ♦♥sér♦♥s A♠♦ ♥

♥ré ♣r ΩA ∪ duiui, i = 1, ..., p, ♦ù ΩA és♥ A♠♦ s ér♥ts

är A. ♦s ♥♦t♦♥s ΩA(log I) t ♣♣♦♥s ♠♦ s ér♥ts

är ♦rt♠qs ♦♥ I. ♦s r♣♣♦♥s q♥ ért♦♥ δ sr Ast t ♦rt♠q ♦♥ I s δ(I) ⊂ I. ♥ és♥ ♣r DerA(log I) A

♠♦ s ért♦♥s ♦rt♠qs ♦♥ I. Pr ♦♥strt♦♥ DerA(log I)st ♥ s♦s èr DerA. ♦s ♦♥sér♦♥s s♦s ♠♦ DerA(log I) DerA(log I) ♦r♠é s δ ts q δ(ui) ∈ uiA ♣♦r t♦s ui ∈ S. ♦s ♣

♣♦♥s ♠♦ s ért♦♥s ♦rt♠qs ♣r♥♣s ♦♥ I. ♦s ♠♦♥

tr♦♥s ♠♠ q DerA(log I) st ΩA(log I).

trtrs P♦ss♦♥ ♦rt♠qs t ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

❯♥ ♦s s ér♥ts ♦r♠s ♦rt♠qs ♦♥strts ♥♦s ♥tr♦s♦♥s

s strtrs P♦ss♦♥ ♦rt♠qs P♦r ♥♦s r♣♣♦♥s q♥ strtr

♦s②♠♣tq sr ♥ rété ♦♠♣① X ♠♥s♦♥ 2n st ♦♥♥é ♥

st♦♥ ω Ω2X(logD) s s r♠s ♦r♠ ér♥t ♦rt♠q

♦♥ ♥ sr rét D X stss♥t s ♣r♦♣rétés s♥ts

ω st r♠é

ωn = ω ∧ ... ∧ ω 6= 0 ♥s H0(X,Ω2n([D])).

♦♥t♦♥ ♠♦♥tr q ♣♦r t♦t r♠ f ♦♥t♦♥ ♦♦♠♦r♣ sr X,

①st ♥ ♥q ♠♣ tr ♦rt♠q δf t q ıδfω = df. ♥ ♦♥sèr

♦rs r♦t f, gω = ω(δf , δg).

s ♣r♦♣rétés t ♦♥ ♠♦♥tr q −,−ω st ♥ strtr P♦s

s♦♥ ♦♦♠♦r♣ ♦rt♠q ♦♥ D. ♦s rst♦♥s ♥♦t♦♥ èr

♥rt t ♥tr♦s♦♥s èr ♥rt ♦rt♠q ♥ r

♥ strtr èr ♥rt ρ : L→ DerA st t ♦rt♠q ♦♥

I s ρ(L) st ♥ s♦s ♠♦ DerA(log I). ♦s ♠♦♥tr♦♥s q t♦t strtr

P♦ss♦♥ ♦rt♠q ♣r♥♣ ♥t sr ΩA(log I) ♥ strtr èr ♥rt ♦rt♠q ♣r s s♦♥ ♣♣t♦♥ ♠t♦♥♥♥ P♦r

♥♦s ♦♥strs♦♥s sr ΩA(log I) ♥ strtr èr ♣r♦♦♥♥t

strtr P♦ss♦♥ ♥t sr ΩA. tt strtr s é♥t sr s é♥értrs

ΩA(log I)− ΩA. Pr

[adu

u, bdv

v] =

a

uu, bdv

v+b

va, vdu

u+ abd(

1

uvu, v).

♥ ♦♥strt ♥s ♥ r♣rés♥tt♦♥ ΩA(log I) ♣r s ért♦♥s ♦rt

♠qs ♦♥ I. ♦♦♠♦♦ tt r♣rés♥tt♦♥ s♣♣ ♦♦♠♦♦

s réstts ♣r♥♣①

P♦ss♦♥ ♦rt♠q ♦s ♠♦♥tr♦♥s q tt ♦♦♠♦♦ st s♦♠♦r♣ à

♦♦♠♦♦ ♠ ♦rt♠q q st s♦♠♦r♣ à ♦♦♠♦♦

P♦ss♦♥ ss♦é ♦rsq strtr P♦ss♦♥ ♦rt♠q ♦♥séré é♦

♥ strtr ♦s②♠♣tq ♠♦②♥ qqs ①♠♣s ♥♦s ♠♦♥tr♦♥s

q♥ é♥ér s ♦♦♠♦♦s P♦ss♦♥ t s P♦ss♦♥ ♦rt♠qs s♦♥t

ér♥ts

Préq♥tt♦♥ ♦rt♠q

♦t ♦r ♥♦s r♠♣ç♦♥s ♥s sé♠ ♣réq♥tt♦♥ r

rété s②♠♣tq ♣r ♥ rété ♦s②♠♣tq (X,D, ω) ♥♦s ♣♦ss à

étr ①t♥s♦♥ s HωX s r♠s ♠♣s trs ♦rt♠qs

♦♠♥t ♠t♦♥♥s rt♠♥t à ω. ♦s r♠♣ç♦♥s ①è♠ ♥

r♠♠ ♣r

0 // CX // (OX , ω) // HωX

// 0

t ♣r♠èr ♣r

0 // OXm // +1 (logD)

σ // DerX(logD) // 0

♦ù +1 (logD) és♥ s r♠s ♦♣értrs ér♥t ♦rt♠qs

♦♥ D. ♥ ♦♥sr♥t ♦r♠ ♥♦s ♥♦s sr♦♥s ♦♦♠♦♦

♠ ♦rt♠q ♣♦r étr ♥térté s ♦r♠s ♦s②♠♣tqs ♦s

♥tr♦s♦♥s ♥♦t♦♥ ért♦♥ ♦♥trr♥t ♦rt♠q à q

♥♦s é♥ss♦♥s ♥♦t♦♥ ss r♥ P♦ss♦♥ ♦rt♠q ♦s ♥♦s ♥

sr♦♥s ♣♦r ♥tr♦r ♥♦t♦♥ ♣réq♥tt♦♥ ♦rt♠q ♦s é♠♦♥

tr♦♥s ♥ té♦rè♠ ♥térté s strtrs P♦ss♦♥ ♦rt♠q ♠♦②♥

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q ss♦é ♥ r s (X,D,Υ) st ♥ r

été P♦ss♦♥ ♦♦♠♦r♣ ♦rt♠q p : L→ X ♥ ré ♥ r♦ts ♦♠♣①s

sr X t Γ(L) s♦♥ ♠♦ st♦♥s ♥ ért♦♥ ♦rt♠q ♦♥trr♥t

Dlog sr p : L → X st ♥ ♣♣t♦♥ C♥ér Ω1X(logD) → EndC(Γ(L)) t

q

Dlogα (fs) = fDlog

α s+ (H(α)f)s

♣♦r t♦t α ∈ Ω1X(logD) t s ∈ Γ(L). ♦s r♠rq♦♥s q s ∇ st ♥ ♦♥♥①♦♥

♦rt♠q sr p : L → X, ♦rs Dα = ∇H(α) st ♥ ért♦♥ ♦♥trr♥t

♦rt♠q sr p : L→ X

s réstts ♣r♥♣①

Prés♥t♦♥s à ♣rés♥t s réstts ss♥ts tt ès

♦t ♦r ♦♥sér♥t sr X = C2 ♦r♠ ω =dy

x♠ér♦♠♦r♣ ♦♥

sr D = 2Y ♦ù Y = (0, y), y ∈ C ♥♦s ♠♦♥tr♦♥s ♥éssté ♠♣♦sr

♣tr ♥tr♦t♦♥ é♥ér

♦♠♠ ②♣♦tès s♣♣é♠♥tr ♦rè♠ ❬t♦ ❪ ♦♥t♦♥ s♦♥

q ♦♥t♦♥ é♥t♦♥ sr ♦t êtr à rré r

♥t s rss ♣rts tr ♥♦s ♦♥s ♦t♥ s réstts s♥ts

♦♥strt♦♥ s èrs P♦ss♦♥ ♦rt♠qs

P♦r t♦t é ♣r♦♣r I ♥ èr ♦♠♠tt ♥tr A ♥♥ré ♣r

S = u1, ..., up ♥♦s ♣♦s♦♥s

DerA(log I) = δ ∈ DerA(log I)δ(ui) ∈ uiA.

st ♠♦ s ért♦♥s ♦rt♠qs ♣r♥♣s ♦♥ I. ♥ ♠♦♥tr

♣tr ♠♠ q

♠♠ DerA(log I) st ΩA(log I).

Pr rs ♥♦s s♦♥s q t♦t strtr P♦ss♦♥ −,− ♦rt♠q sr

A ♦♥ I ♥t ♥ ♣♣t♦♥ H : ΩA → DerA é♥ ♣r H(df) = f,−♣♣é ♣♣t♦♥ ♠t♦♥♥♥ q st ♥ ♦♠♦♠♦r♣s♠ A♠♦s

♣s ♦♥ ♠♦♥tr ♦r ♠♠ q

♠♠ ♣♣t♦♥ ♠t♦♥♥♥ H ss♦é à ♥ strtr P♦ss♦♥

♦rt♠q st à ♠ ♥s DerA(log I).

♥ ♥ ét ss ♠♠ s♥t

♠♠ ♦t S = u1, ...up ♥ st éé♠♥ts A rt♠♥t ♣r

♠èr (ui) 6= (uj) t ui /∈ (uj), uj /∈ (ui) ♣♦r t♦t i 6= j. ♦t −,− ♥

strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I = 〈S〉A.♦rs

1

uiui,− ∈ DerA(log I) t

1

uiujui, uj ∈ A.

♥ ♥ ét q

♦r♦r −,− st ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣

♦♥ ♥ é I ♥♥ré ♣r ♥ st ♥ éé♠♥ts A rt♠♥t ♣r♠èr

♦rs ♣♣t♦♥ ♠t♦♥♥♥ ss♦é H s ♣r♦♦♥ ♥ ♥ ♦♠♦♠♦r♣s♠

A♠♦s

H : ΩA(log I) → DerA(log I).

♥ ♠♦♥tr ♥ ♣s q H st ♥ ♦♠♦♠♦r♣s♠ èr ♦rsq♦♥ éq♣

ΩA(log I) r♦t é♥ ♠♠

s réstts ♣r♥♣①

♦♥strt♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q tqqs ①♠♣s

♦♥strt♦♥ tt ♦♦♠♦♦ r♣♦sé sr té♦rè♠ s♥t

é♦rè♠ ♦t strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ ♥

é I ♥ Rèr A ♥t sr ΩA(log I) ♥ strtr ♥rt

tr♠♥t t ♣♦r t♦t strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ ♥

é I, (ΩA(log I), H, [−,−]) st ♥ èr ♥rt

♦s és♦♥s té♦rè♠ q t♦t strtr P♦ss♦♥ ♦rt♠q ♥t

♥ r♣rés♥tt♦♥ ΩA(log I) ♣r s ért♦♥s ♦rt♠qs ♦s ♣♣♦♥s

♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q ♦♦♠♦♦ ss♦é à tt r♣rés♥

tt♦♥ ♦s ♥♦t♦♥s HkPS kè♠r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

ttr P t réér♥ à P♦ss♦♥ ♦rs q ttr S t réér♥ à t♦

♦s és♥♦♥s ♣r HkP kè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦s ♠♦♥tr♦♥s

q strtr P♦ss♦♥ é♥ ♣r x, y = x st ♦rt♠q ♣r♥♣

♦♥ é xC[x, y] t ss r♦♣s ♦♦♠♦♦s s♦♥t

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦ P♦ss♦♥ x, y = x s♦♥t

H0P∼= C H1

P∼= C t H2

P∼= 0A.

♥ ♠♦♥tr ss q ss r♦♣s ♦♦♠♦♦s P♦ss♦♥ ♦rt♠qs s♦♥t

H0PS

∼= C, H1PS

∼= C t H2PS

∼= 0A.

♥ r♠rq q s ① r♦♣s s♦♥t s♦♠♦r♣s st û t q str

tr P♦ss♦♥ x, y = x st ♦s②♠♣tq ♦r♠ ♦s②♠♣tq ss♦é

ω0 =dx

x∧ dy.

Pr rs ♥♦s ♠♦♥tr♦♥s q x, y = x2 é♥t ♥ strtr P♦ss♦♥ ♦

rt♠q ♦♥ x2C[x, y] q ♥st ♣s ♦s②♠♣tq r ♦r♠

ss♦é stdx

x2∧ dy q ♥st ♣s ♦rt♠q ♦♥ x2C[x, y]. Pr ♦♥tr

♥♦s ♠♦♥tr♦♥s q ss r♦♣s ♦♦♠♦♦s P♦ss♦♥ t P♦ss♦♥ ♦rt

♠q s♦♥t s♦♠♦r♣s t ♦♥♥és ♣r

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

x, y = x2 s♦♥t

H1PS

∼= C[y]⊕ C1[x];H2PS

∼= C[y], H0PS

∼= C.

s♥st q t êtr ♦s②♠♣tq ♥st ♣s ♥ ♦♥t♦♥ ♥éssr é

té ♥tr s ① r♦♣s ♦♦♠♦♦

♦s ♠♦♥tr♦♥s ss q strtr P♦ss♦♥ (x, y = 0, x, z = 0, y, z =

xyz) ♥s A = C[x, y, z] st ♦rt♠q ♦♥ xyzC[x, y, z] t q s♦♥

tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q st ♥ s♦s r♦♣

s♦♥ tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥

♣tr ♥tr♦t♦♥ é♥ér

é♦rè♠ tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt

♠q strtr P♦ss♦♥ (A = C[x, y, z], x, y = 0, x, z =

0, y, z = xyz) st

H3P∼= C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xyC[x]⊕xzC[x]⊕ xzC[z]⊕ yzC[y]⊕ yzC[z],

tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦s♦♥ strtr P♦ss♦♥ (A =

C[x, y, z], x, y = 0, x, z = 0, y, z = xyz) st

H3PS

∼= C[y]⊕ zC[z]⊕ xC[x].

♥ s♦♠♠ ♥♦s ♣♦♦♥s ♦♥r q♥ é♥ér s r♦♣s ♦♦♠♦♦

P♦ss♦♥ ♦rt♠q s♦♥t ♥♦♥ tr① t st♥ts ① P♦ss♦♥ ss♦és

r ♥♦♥ trté r ♣r♠t ♥s ♥ rt♥ ♠sr ♦r rô ss

♥t ♥r♥ts

♦s ♠♦♥tr♦♥s q t♦t strtr ♦s②♠♣tq é♥t ♥ t r

été ♥ s s s②♠♣tqs ♠♥s♦♥ ♥

♣♣t♦♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♣rès ♥ ét ♦♥t♦♥ ♥térté s ♦r♠s ér♥ts ♦rt

♠qs ♥♦s ♥♦s sr♦♥s ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q ♣♦r étr

♣réq♥tt♦♥ s strtrs P♦ss♦♥ ♦s②♠♣tqs ♦s ♣r♦♣♦s♦♥s s

réstts s♥ts

é♦rè♠ ♦t ω ♥ 2♦r♠ ♦rt♠q r♠é ♦♥ ♥ sr

rét D ♥ rété ♦♠♣① X ♠♥s♦♥ 2n. D stst ♣r♦♣rété

é♦rè♠ ♦rs s ♣r♦♣rétés s♥ts s♦♥t éq♥ts

ω =dh

h∧ ψ + η st ♥tér

res(ω) st ①t t ①st [ω0] ∈ H2(X,C) ♥tér t q

[ω0] = [η].

♦s s ②♣♦tèss

D st à r♦s♠♥t ♥♦r♠①

D = ∪j∈IDj st é♦♠♣♦st♦♥ ♥ ♦♠♣♦s♥ts rréts D ♦rs

q Dj st ss I és♥♥t ♥ ♥s♠ ♥s

♦♥sérés ♥s ❬♦t♦ ❪ ♣♦r rtérst♦♥ s sss r♥ ♦rt

♠qs ♥♦s ♠♦♥tr♦♥s q s ∂D és♥ ér♥t P♦ss♦♥ ♦rt♠q

♦rs ♦♥

Pr♦♣♦st♦♥ ♦t (X,D,Υ) ♥ rété P♦ss♦♥ ♦rt♠q ♦♥

♥ sr D stss♥t s ②♣♦tèss t (X,D,Υ) st ♦ ♣réq♥

t s ①st ♥ ♠♣ trs ♦rt♠q δ t ♥ ♦r♠ ♦rt♠q

ω ♥tér t q

Υ+ ∂Dδ = H(ω).

♦tés

♥ r♥♥t à rtérst♦♥ s ♦r♠s ♦rt♠qs ♥térs ♥♦s ♣r♦♦♥s

q

♦r♦r ♦t (X,D,Υ) ♥ rété P♦ss♦♥ ♦rt♠q ♦♥

♥ sr D stss♥t s ②♣♦tèss é♦rè♠ (X,D,Υ) st

♦ ♣réq♥t s ①st ♥ ♠♣ trs ♦rt♠q δ s ♦♥t♦♥s

♦♦♠♦r♣s Ri, i = 1, ..., k t ♥ ♦r♠ ω0 ♦♦♠♦r♣ sr ♥ s♦s rété

♠♥s♦♥ ♥ X ♥tér t q

Υ+ ∂D(δ −k∑

i=1

Ri

hi(H(dhi)) = H(ω0).

♦tés

r s ♦r♠s ér♥ts ♦rt♠qs

♦s ♦♥s s♦♥é à s♦s st♦♥ ♥éssté ♠♣♦sr ♦♥t♦♥

s♦♥ q ♦♥t♦♥ é♥t♦♥ sr ♦t êtr à rré r

♦s ♦♥s ss ♦♥strt ♠♦ s ér♥ts ♦r♠s ♦rt♠qs

♦♥ ♥ é I t ♥♦s ♦♥s rtérsé s♦♥ ♠♦ ♦rsq I st ♥♥ré

♣r ♥ ♥♦♠r ♥ éé♠♥ts èr s♦s♥t

r s strtrs P♦ss♦♥ ♦rt♠qs t ♥r♥ts♦♦♠♦♦qs

tr♠ tt ét ♥♦s s♦♠♠s ♣r♥ à ♠ttr sr ♣ té♦r

P♦ss♦♥ ♦rt♠q ♣s ♥♦s ♦♥s ♠♦♥trr q ♥s s s strtrs

P♦ss♦♥ ♦rt♠qs ♣r♥♣s ♣♣t♦♥ ♠t♦♥♥♥ ss♦é st à ♠

♥s ♠♦ s ért♦♥s ♦rt♠qs t q s ♣r♦♦♥ sr ♠♦

s ér♥ts ♦r♠s ♦rt♠qs ♥s s ♥♦s ♦♥s ♦♥strt sr

♠♦ s ér♥ts ♦r♠s ♦rt♠qs ♥ r♦t ♣r♦♦♥♥t

P♦ss♦♥ ♥t sr ♠♦ s ér♥ts ♦r♠s ♦s ♦♥s

①♣♦té s ♣r♦♣rétés ♣♦r érr ♣♣t♦♥ ♠t♦♥♥♥ ♥t ♣r s str

trs P♦ss♦♥ ♦rt♠qs ♥ strtr èr ♥rt sr ♠♦

s ér♥ts ♦r♠s ♦rt♠qs râ à tt r♥èr ♥♦s ♦♥strs♦♥s

♦♠♣① P♦ss♦♥ ♦rt♠q t ♦♥s qqs r♦♣s ♦♦♠♦♦

s ss♦és ♦s ♦♥s ♠♦♥tré sr s ①♠♣s q s r♦♣s ♦♦♠♦♦

s s♦♥t ♥ é♥érs ér♥ts s r♦♣s ♦♦♠♦♦s P♦ss♦♥ ss♦és

♥ qs ♦ï♥♥t ♦rsq strtr P♦ss♦♥ st ♥t ♣r ♥ strtr

♦s②♠♣tq râ à tt ♥♦ ♦♦♠♦♦ ♥♦s ♦♥s ♥tr♦t ♥♦t♦♥

♣réq♥tt♦♥ ♦rt♠q q ♥♦s ♦♥s é♠♦♥tré ♥ réstt

♣réq♥tt♦♥ ts strtrs ♥♦t♦♥ ért♦♥ ♦♥trr♥t

♦rt♠q

♣tr ♥tr♦t♦♥ é♥ér

Prs♣ts

r s èrs P♦ss♦♥ ♦rt♠qs

♦s ♦♥s ♥tr♦t ♥♦t♦♥ rété P♦ss♦♥ ♦rt♠q t ♦

♦♠♦♦ P♦ss♦♥ ♦rt♠q sr ♥térss♥t étr s ♣r♦♣rétés

érqs s èrs P♦ss♦♥ ♦rt♠q ♥ r♠♣ç♥t sr ♣r ♥

é q♦♥q ♥ èr ss♦t ♦♥♥é ♦s ♦♥s ♦♠♠♥é tt ét

♣tr ♠s ♥♦s ♥♦s s♦♠♠s ♠tés s s strtrs P♦ss♦♥ ♦

rt♠qs ♣r♥♣s sr ♣rtèr♠♥t ♥térss♥t étr s é♥ér

s strtrs P♦ss♦♥ ♦♥t r♦t st ♥ ért♦♥ ♦rt♠q Pr

rs ♦s ♦♥s ♦♥strt ♥s s ♦ù é I st ♥♥ré ♣r ♥ st

♥ éé♠♥ts èr ♠♦ s ér♥ts ♦r♠s ♦

rt♠qs ♥ ♣♦rr rrr s é♥ér ♦ù I st ♥ é q♦♥q ♥

♣♦rr ss rrr ♥♦ ♦rt♠q té♦rè♠ ♦ss♦st♥t

♦s♥r ♥s ❬♦ss t ❪

r s ♦r♠s ér♥ts ♦rt♠qs

qst♦♥ q ♦♥ ♥ ♣t s ♣r♠ttr ♦r st s♦r ♦♠♠♥t

s♦♥t s ♦r♠s ér♥ts ♦rt♠qs ♦♥ s srs ♥♦♥ rét ♥

♣♦rr ♦♥ ♥s ♥ ♣r♦ ♥r s ♣♥r sr st ♣♦r ♦t ♥

♣rt rr s réstts ♥♦s é♦rè♠ ♦♠♣rs♦♥ ♦rt

♠q tr ♣rt rtérst♦♥ s s① s r♠s ♦r♠s ér♥

ts ♦rt♠qs ss♦és t ♥♥ q♥ té♦rè♠ rtérs♥t s srs

♣♦r sqs s s① s♦♥t rs

r q♥tt♦♥

q♥tt♦♥ é♦♠étrq st ssé ♥ ① r♥s ét♣s ♣r♠èr

ét♥t ♣réq♥tt♦♥ ♦s ♥♦s s♦♠♠s ♣s ♦sés sr tt r♥èr ét♣

ét♣ ♣♦rst♦♥ ♠♦②♥ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q rst

♥①♣♦ré Pr rs srt ♥térss♥t rrr ♠♣t ♦♦♠♦♦

P♦ss♦♥ ♦rt♠q sr q♥tt♦♥ ♣r é♦r♠t♦♥

♣tr

r s strtrs P♦ss♦♥

♦rt♠qs

♦♠♠r èrs P♦ss♦♥ ♦rt♠qs

ért♦♥s ♦rt♠qs

ér♥ts ♦r♠s ♦rt♠qs

éré ♣r r♣♣♦rt à ♥ ért♦♥ ♦rt♠q

trtrs èrs P♦ss♦♥ ♦rt♠qs

qs ①♠♣s èrs P♦ss♦♥ ♦rt♠qs

❱rétés P♦ss♦♥ ♦rt♠qs

sr r

❯♥ r♠rq sr s ♦r♠s ér♥ts ♦rt♠qs t

♠♣s trs ♦rt♠qs

é♥t♦♥ t ♣r♠èrs ♣r♦♣rétés

❱rétés ♦s②♠♣tqs

qs ①♠♣s rétés P♦ss♦♥ ♦rt♠qs

s♣ s SU(2) ♠♦♥♦♣ôs ♠♥étqs r

♥tr♦t♦♥

♣tr st ♦♥sré à ♦♥strt♦♥ èrs P♦ss♦♥ ♦rt♠qs

t rétés P♦ss♦♥ ♦rt♠qs P♦r ♥♦s ♦♥strs♦♥s ♠♦ s

ér♥ts ♦r♠s ♦rt♠qs t ét♦♥s qqs ♥s ss ♣r♦♣rétés

èrs P♦ss♦♥ ♦rt♠qs

♥s tt ♣rt ♦♥ és♥r ♣r

• A ♥ èr ss♦t ♦♠♠tt ♥tr t ♥tèr sr ♥ ♦r♣s k

rtérstq

• I ♥ é A,• U r♦♣ ♠t♣t s ♥tés A.• DerA A♠♦ s ért♦♥s A.• ΩA ♠♦ s ér♥ts är A

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

ért♦♥s ♦rt♠qs

é♥t♦♥ ♥ ♣♣ ért♦♥ ♦rt♠q ♦♥ ♥ é I A t♦t

éé♠♥t D DerA t q D(I) ⊂ I.

♥ ♥♦tr DerA(log I) ♥s♠ s ért♦♥s A ♦rt♠qs ♦♥ IP♦r t♦s D1, D2 ∈ DerA(log I) ♦♥

D1(D2)(I) ⊂ I.

♥ ♥ ét q DerA(log I) st st ♣♦r r♦t s ért♦♥s

A.

♠♠ DerA(log I) st ♥ s♦s èr DerA.

♦t S = u1, ..., up ♥ s♦s ♥s♠ à p éé♠♥ts A− U .♥ s♣♣♦s ♥s st q I st ♥♥ré ♣r S t ♦♥ és♥r ♣r uA é

A ♥♥ré ♣r u ∈ S.

é♥t♦♥ S st t rt♠♥t ♣r♠r s s é① uiA t ujA s♦♥t ♣r♠rs

t ♥trst♦♥ tr ♣♦r t♦t i, j ts q i 6= j

P♦s♦♥sDerA(log I) := D ∈ DerA(log I);D(ui) ∈ uiA; ♣♦r t♦t ui ∈ S.

P♦r t♦s D ∈ DerA(log I) t ui ∈ S, ♦♥

D(ui)

ui∈ A.

Pr rs ♣♦r t♦s D1, D2 ∈ DerA(log I) ♦♥

D1(D2(ui)) = D1(uib2) ♦ù D2(ui) = uib2= uiD1(b2) + b2D1(ui)

= ui(D1(b2) + b2b1) ♦ù D1(ui) = uib1

P♦r t♦t i = 1, ..., p. ♦♥ DerA(log I) st st ♣♦r r♦t s ér

t♦♥s A. ♦ù ♠♠ s♥t

♠♠ DerA(log I) st ♥ s♦s èr DerA(log I).

é♥t♦♥ s éé♠♥ts DerA(log I) s♦♥t ♣♣és ért♦♥s ♦rt♠qs

♣r♥♣s ♦♥ I.

ér♥ts ♦r♠s ♦rt♠qs

P♦s♦♥s MA té♦r s A♠♦s

Pr♦♣♦st♦♥ ♥♦♦♥tr Der(A,−) MA st r♣rés♥t

èrs P♦ss♦♥ ♦rt♠qs

Pr ♣rès ♣r♦♣rété ♥rs ♣r♦t t♥s♦r A⊗A ♣♣t♦♥

m : A⊗A → A(a, b) 7→ ab

♥t ♣r ♣r♦t A, st ♥ ♦♠♦♠♦r♣s♠ kèrs ♥ ♣♦s ker(m) =

I t ♦♥ ♠♦♥tr q I st ♥As♦s ♠♦ A⊗A ♥♥ré ♣r a⊗1A−1A⊗a, a ∈A. s ♠♦s q♦t♥ts B = A⊗A/I2 t ΩA = I/I2 s♦♥t ♦♥ ♥ é♥s

tr ♣rt a = m(1⊗a) ♣♦r t♦t a ∈ A. ♦♥ m ♥t ♥ s♦♠♦r♣s♠

A ⊗A/I ≃ A q à s♦♥ t♦r ♥t ♥ é♣♠♦r♣s♠ m : B → A → 0. ♥ ♦♥

st ①t ♦rt s♥t

0 → ΩA → B → A→ 0.

Pr rs s ♠♦r♣s♠s

λ1 : A → B, a 7→ a⊗ 1 + I2λ2 : A → B, a 7→ 1⊗ a+ I2

ér♥t s étés s♥ts mλ1 = mλ2 = 1A. s♦♥t ♦♥ s st♦♥s tt

①t♥s♦♥ ♣s s étés mλ1 = mλ2 = 1AA s♥st q λ1 t λ2 s♦♥t s

rè♠♥ts 1A ♦♥ λ1 − λ2 = d st ♥ éé♠♥t Der(A,ΩA)

P♦r r ♣r ♥♦s ♦♥s ♠♦♥trr q (ΩA, d) st ♥rs

♦t D ∈ Der(A,M) ♣♣t♦♥ ϕ : A⊗A →M ⊕A

x⊗ y 7→ (xy, xDy) = (m(x⊗ y), xDy)

st ♥ ♦♠♦♠♦r♣s♠ kèrs q st A♥ér

Psq m(∑xi ⊗ yi) =

∑xiyi = 0 ♣♦r t♦t

∑xi ⊗ yi ∈ I, ♦rs rstrt♦♥

ϕ : ϕ |I : I −→ M st A♥ér ♣s t q ϕ(I2) = 0 ♠♣q I2 ⊂ ker ϕ.

s♥st q ϕ ♥t ♥ ♦♠♦♠♦r♣s♠ f : ΩA −→ M t q f π = ϕ ♦ù π

és♥ ♣r♦t♦♥ ♥♦♥q I sr I/I2. Pr rs ♣♦r t♦t a ∈ A, ♦♥

f(da) = f(1⊗ a− a⊗ 1 + I2)

= ϕ(1⊗ a− a⊗ 1)

= Da

è é♠♦♥strt♦♥

é♥t♦♥ ΩA st ♣♣é ♠♦ s ér♥ts ♦r♠s

Pr♦♣♦st♦♥ ♠♦♥tr q ♣♦r t♦t A♠♦ M, ①st ♥ s♦♠♦r

♣s♠ σM : Hom(ΩA,M) ∼= Der(A,M). ♥s st s♦♠♦r♣s♠ σA :

Hom(ΩA,A) ∼= Der(A) sr ♥♦té σ.

ér♥ts ♦rt♠qs

é♥t♦♥ ♥ ♣♣ ♠♦ s ♦r♠s ér♥ts ♦rt♠qs ♦♥

I A♠♦ DerA(log I).

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

♥ ♣♦s

ΩA(log I)

A♠♦ ♥♥ré ♣r

duiui, ui ∈ S, i = 1, ..., p

∪ ΩA.

é♥t♦♥ ΩA(log I) st ♣♣é ♠♦ s ér♥ts ♦r♠s A ♦

rt♠qs ♦♥ I.

♦t δ ∈ DerA(log I). ♣rès ♣r♦♣♦st♦♥ δ ♥t ♥ ♣♣t♦♥ A

♥ér

σ(δ) : ΩA → A

t q

σ(δ) d = δ.

♦♥ ♣♦r t♦t u ∈ S, ①st ϕ(u) ∈ A t q

σ(δ)(du) = δu = uϕ(u)

t ♦♥1

uσ(δ)(du) = ϕ(u) ∈ A.

♥ ♣♣t♦♥ ♥ér s♥t

σ(δ) : ΩA(log I) −→ Aadu

u+ bdf 7→ a

1

uσ(δ)(du) + bσ(δ)(df).

ér σ(δ1 + gδ2)(adu

u+ bdf) = σ(δ1)(a

du

u+ bdf) + gσ(δ2)(

du

u+ bdf).

♥t ♦♥ ♥ ♣♣t♦♥ A♥ér

σ : DerA(log I) −→ HomA(ΩA(log I),A)

δ 7→ σ(δ) : adu

u+ bdf 7→ a

1

uσ(δ)(du) + bσ(δ)(df)

q st ♥ s♦♠♦r♣s♠

♥ t ♣♦r t♦t f ∈ HomA(ΩA(log I),A) ♦♥ f d ∈ DerA(log I).♥ ♦♥sèr ♦♠♦♠♦r♣s♠ A♠♦s

ψ : HomA(ΩA(log I),A) → DerA(log I)f 7→ f d.

P♦r t♦t δ ∈ DerA(log I) ♦♥

ψ σ(δ) = ψ(σ(δ))

= σ(δ) d= δ.

èrs P♦ss♦♥ ♦rt♠qs

♦♥ ψ σ = id DerA(log I).

♠ê♠ ♣♦r t♦s f ∈ HomA(ΩA(log I),A) t u ∈ S ♦♥

[(σ ψ)(f)](du) = [σ(ψ(f))](du)

= [σ(f d)](du)= σ(f d)(du)= (f d)u= f(du)

Pr rs[(σ ψ)(f)](duu ) = σ(f d)(duu )

=1

u(σ(f d)(du))

=1

uσ(f d) d(u)

=1

u(f d)(u)

= f(duu ).

♣♦r t♦t u ∈ S.♦♥

(σ ψ)(f) = f ♣♦r t♦t f ∈ HomA(ΩA(log I),A). t ♦♥ σ ψ =

idHomA(ΩA(log I),A). ♣r♦ ♠♠ s♥t

♠♠ DerA(log I) st ΩA(log I).P♦r t♦s fda ∈ ΩA t δ ∈ DerA ♦♥

σ(δ)(fda) = f(σ(δ) d)(a)= fδ(a).

♥ ♦♥sèr ♣♣t♦♥

θ : ΩA → Hom(DerA,A)

ω 7→ θ(ω) : δ 7→ σ(δ)(ω)

θ st ♣r ♦♥strt♦♥ ♥ ♦♠♦♠♦r♣s♠ A♠♦s

Pr rs ♣♦r t♦s δ ∈ DerA(log I) t u ∈ Sθ(du)δ = σ(δ)(du)

= (σ(δ) d)u= δ(u) ∈ uA.

♦♥1

uθ(du)(δ) ∈ A ♣♦r t♦t δ ∈ DerA(log I) t u ∈ S.

θ ♥t ♦♥ ♥ ♦♠♦♠♦r♣s♠ A♠♦s

Θ : ΩA(log I) −→ Hom( DerA(log I),A)

fdu

u+ gda 7→ f

uθ(du) + gθ(da)

q s ♣r♦♦♥ ♥ ♦♠♦♠♦r♣s♠ èrs rés

Θ :∧ΩA(log I) −→ Lalt( DerA(log I),A)

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

éré ♣r r♣♣♦rt à ♥ ért♦♥ ♦rt♠q

♥ ♥♦t∧

A[ΩA(log I)] :=⊕n∈N

∧nA[ΩA(log I)] Aèr ①térr A

♠♦ ΩA(log I). ért♦♥

d : A → ΩA

♦♥ ét ért♦♥

d : A → ΩA(log I) a 7→

da s a ∈ A− I∗

ada

as a ∈ I∗

d s ♣r♦♦♥ ♥ ♥ ért♦♥ éré

d :∧

A

[ΩA(log I)] →∧

A

[ΩA(log I)]

t q ♦♣ (∧

A[ΩA(log I)], d) s♦t ♥ ♦♠♣① ér♥t

P♦r t♦t δ ∈ DerA(log I) t t♦t ♥tr p ≥ 1 ♣♣t♦♥

σδ : [ΩA(log I)]p →∧

A

[ΩA(log I)], (ω1, ..., ωp) 7→p∑

i=1

(−1)i−1σ(δ)(ωi)ω1∧ω2∧...∧ωi∧...ωp

st A♠t♥ér tr♥é ♥ ♥♦t

iδ :∧

A

[ΩA(log I)] →∧

A

[ΩA(log I)]

♥q ♣♣t♦♥ A♥ér t q

iδ(ω1 ∧ ω2 ∧ ... ∧ ωp) 7→p∑

i=1

(−1)i−1σ(δ)(ωi)ω1 ∧ ω2 ∧ ... ∧ ωi ∧ ...ωp

♣♦r t♦t p

♠♠ ♣♣t♦♥ iδ st ♥ ért♦♥ ré −1

Pr ♣r st s♠♣ t rt

é♥t♦♥ ♦♣értr ré ③ér♦ Lδ := iδ d + d iδ st ♣♣é éré

♣r r♣♣♦rt à ért♦♥ ♦rt♠q δ.

♣r♦♣♦st♦♥ s♥t ♦♥♥ qqs ♣r♦♣rétés Lδ

Pr♦♣♦st♦♥ P♦r t♦t δ ∈ DerA(log I), ω ∈ ΩA(log I) t a ∈ A, ♦♥

Laδ(ω) = aLδ(ω) + (σ(δ))(ω)d(a)

Lδ(aω) = δ(a).ω + aLδω Lδ(d(a)) = d[δ(a)]

èrs P♦ss♦♥ ♦rt♠qs

Pr

P♦r t♦t a ∈ A t ω ∈ ΩA(log I), ♦♥

Laδ(ω)= iaδ(d(ω)) + d(iaδ(ω))

= aiδ(d(ω)) + d(aiδ(ω))

= aiδ(d(ω)) + iδ(ω)d(a) + ad(iδ(ω))

= aLδω + σ(δ)(ω)d(a)

P♦r t♦s a ∈ A t ω ∈ ΩA(log I), ♦♥

Lδ(aω)= iδ(d(aω)) + d(iδ(aω))

= iδ(ad(ω) + da ∧ ω) + d(aiδ(ω))

= aiδ(d(ω)) + iδ(d(a) ∧ ω) + ad(iδω) + iδ(ω)d(a)

= aiδ(d(ω)) + σ(δ)(d(a))ω − σ(δ)(ω)d(a) + ad(iδ(ω)) + σ(δ)(ω)d(a)

= aLδω + σ(δ)d(a)ω

P♦r t♦t a ∈ A, ♦♥

Lδ(d(a)= iδ(d(d(a))) + d(iδ(d(a)))

= d(iδ(d(a)))

= d(σ(δ) d(a))

trtrs èrs P♦ss♦♥ ♦rt♠qs

trtrs èrs P♦ss♦♥

❯♥ èr P♦ss♦♥ st ♥ èr ss♦t A ♠♥ ♥ ♣♣t♦♥

♥ér −,− ♥ts②♠étrq ér♥t s ① ♣r♦♣rétés s♥ts

a, b, c+ b, c, a+ c, a, b = 0 ♥tté ♦

a, bc = ba, c+ ca, b ♣r♦♣rété ♥③

♥ ♥ ét ♦♥ q ♣♦r t♦t a ∈ A ♣♣t♦♥

ada : A → A, b 7→ a, b

st ♥ ért♦♥ sr A. ♣s ♣♦r t♦s a, b ∈ A, ♦♥

adab(x) = ab, x = ab, x+ ba, x

s♥st q ♣♣t♦♥ ad : A → DerA, a 7→ ada st ♥ ért♦♥ sr

A à rs ♥s A♠♦ DerA. ♥t ♣rès ♣r♦♣♦st♦♥ ♥

♦♠♦♠♦r♣s♠ A♠♦s

H : ΩA → DerA

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

t q H d = ad.

H st ♣♣é ♣♣t♦♥ ♠t♦♥♥♥ ss♦é à −,−.♥ ♥ ét ♥ ♦♠♦♠♦r♣s♠ A♠♦s

−σ H : ΩA → Hom(ΩA,A)

à ♣rtr q ♦♥ é♥t ♣♣t♦♥

ω : ΩA × ΩA → A, (x, y) 7→ −[(σ H)(x)]y

ω st ♥ ♦r♠ A♥ér tr♥é

♥ t ♣♦r t♦t x =∑

j∈J ;J♥

ajdbj ∈ A, ♦♥

ω(x, x) = −[σ H)(x)](x)

= −∑

j∈J ;J♥

aj [σ(H(x))]dbj

= −∑

j∈J ;J♥

aj [H(x)]bj

= −∑

j∈J ;J♥

aj∑

k∈J ;J♥

ak[H(dbk)]bj

= −∑

j∈J ;J♥

aj∑

k∈J ;J♥

ak[ad(bk)]bj

= − ∑j,k∈J ;J♥

ajakbk, bj

= 0

ω st ♣♣é ♦r♠ P♦ss♦♥ ss♦é à −,−.♦rsq A és♥ èr C∞(M) s ♦♥t♦♥s ér♥ts sr ♥ rété

P♦ss♦♥ M, ω st ♣♣é t♥sr P♦ss♦♥ ♦ tr P♦ss♦♥

Pr rs ♣♦r t♦t a, b ∈ A ♦♥

ω(da, db) = −[σ(H(da))](db)

= −[σ(H(da))] d(b)= −H(da)b

= −a, b

t ♣r stLH(da)db = d(iH(da)db)

= d(H(da)b)

= d(H(da)b)

= da, b

Pr♦♣♦st♦♥ P♦r t♦t èr P♦ss♦♥ A ♦r♠ P♦ss♦♥ ω, ♣

♣t♦♥

[−,−] : ΩA × ΩA −→ ΩA

(x, y) 7→ d(ω(x, y)) + LH(x)y − LH(y)x

é♥t ♥ strtr kèr sr ΩA. ♣s

èrs P♦ss♦♥ ♦rt♠qs

[x, ay] = (H(x))(a)y + a[x, y]

s ♣♣t♦♥s

d : A → ΩA

t

H : ΩA → DerA

s♦♥t s ♦♠♦♠♦r♣s♠s kèrs

s♥st q t♦t strtr P♦ss♦♥ −,− ♥t sr ΩA ♥ strtr

èr ♥rt ♥ ♣rtr ♣♦r x = adu, y = bdv ∈ ΩA ♦♥

♥ ♣rt

d(ω(adu, bdv)) = −d(abu, v) = −au, vdb− bu, vda− abdu, v,

tr ♣rt

LH(adu)bdv = au, bdv + abd(u, v) + bu, vdaLH(bdv)adu = bv, adu+ abd(v, u) + av, udb.

♦♥

[adu, bdv] = −au, vdb− bu, vda− abdu, v+ au, bdv+abd(u, v) + bu, vda− bv, adu− abd(v, u)− av, udb.

t ♣r st

[adu, bdv] = au, bdv + ba, vdu+ abdu, v.

trtr P♦ss♦♥ ♦rt♠q

♣rès q ♣réè t♦t strtr P♦ss♦♥ ♥s A st ♥ ért♦♥

sr A t DerA(log I) st ♥ s♦s èr DerA.

é♥t♦♥ ❯♥ strtr P♦ss♦♥ −,− sr A st t ♦rt♠q ♦♥

I s st ♥ ért♦♥ ♦rt♠q ♦♥ I

♦t −,− ♥ strtr P♦ss♦♥ ♦rt♠q ♦♥ I. P♦r t♦s a ∈ At u ∈ I ♦♥

a, u ∈ A, I ⊂ I♦♥ a,− st ♥ ért♦♥ ♦rt♠q ♦♥ I s♥st ♦♥ q ♣♦r

t♦t a ∈ A,H(da) = a,− ∈ DerA(log I).

♠♠ ♣♣t♦♥ ♠t♦♥♥♥ H t♦t strtr P♦ss♦♥ ♦

rt♠q st à rs ♥s DerA(log I).

é♥t♦♥ ❯♥ strtr P♦ss♦♥ −,− sr A st t ♦rt♠q ♣r♥

♣ ♦♥ I s ♣♦r t♦t u ∈ Su,− ∈ DerA(log I).

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

♦s r♠rq♦♥s q s −,− st ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣

♦♥ I, ♦rs ♣♦r t♦t x ∈ S ①st ♥ ♣♣t♦♥ ϕx : A → A t q

x, a = xϕx(a)

♣♦r t♦t a ∈ A. ♥s s −,− st ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥

♣ ♦♥ I ♦rs ♣♦r t♦s x ∈ S t a, b ∈ A, ♦♥

x, ab = ax, b+ bx, a= x(aϕx(b) + bϕx(a))

t

x, ab = xϕx(ab)

♦♥

ϕx(ab) = aϕx(b) + bϕx(a).

Pr rs ♣♦r t♦s x, y ∈ S ♦♥

xϕx(y) = x, y = −yϕy(x). ♣s I st ♣r♠r ♦rs ①st bxy ∈ I t q

x, y = xybxy.

♥ ♥ ét q

ϕx(ay) ∈ yA.P♦r t♦s a ∈ A t y ∈ S. ♣s I st ♣r♠r ♦rs ϕx ∈ DerA(log I) ♣♦r

t♦t x ∈ S.♥ ♥ ét ♠♠ s♥t

♠♠ ♦t S = u1, ...up ♥ s♦s ♥s♠ A rt♠♥t ♣r♠r

(ui) 6= (uj) t ui /∈ (uj), uj /∈ (ui) ♣♦r t♦t i 6= j. ♦t −,− ♥ strtr

P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I = 〈S〉A.♦rs

1

uiui,− ∈ DerA(log I) t

1

uiujui, uj ∈ A

♥ s♣♣♦s q♥ ♣s s éé♠♥ts S s♦♥t rt♠♥t ♣r♠rs q ♠♣q

q ♣♦r t♦t u ∈ S, 1

uH(du) ∈ DerA(log I). ♥ ♥ ét ♦♠♦♠♦r♣s♠

A♠♦s

H : ΩA(log I) → DerA(log I)é♥ ♣r

ΩA(log I) ∋ x =∑

ui∈S,ai∈A

aiduiui

+∑

vj∈A,i∈J,bj∈A,J♥

bjdvj

_

H∑

ui∈S,ai∈A

aiuiH(dui) +

∑vj∈A,i∈J,bj∈A,J♥

bjH(dvj)

.

♥ H|ΩA= H.

èrs P♦ss♦♥ ♦rt♠qs

♦r♦r −,− st ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥

♣ ♦♥ ♥ é I ♥♥ré ♣r ♥ st ♥ éé♠♥ts A rt♠♥t

♣r♠èr ♦rs ♣♣t♦♥ ♠t♦♥♥♥ ss♦é H s ♣r♦♦♥ ♥ ♥ ♦♠♦♠♦r

♣s♠ A♠♦s

H : ΩA(log I) → DerA(log I).

é♥t♦♥ H st ♣♣é ♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é à

strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ −,−.

♦r♦r t ♣r ♠♠ ♦♥ ét q t♦t strtr

P♦ss♦♥ ♦rt♠q ♣r♥♣ ♥t ♥ ♦♠♦♠♦r♣s♠ A♠♦s

Φ : ΩA(log I) → Hom(ΩA(log I),A)

α 7→ σ H(α)

♥ ♥ ét ♦♥ ♦r♠ s♥t sr ΩA(log I)

π(α, β) := [Φ(x)]y.

Pr♦♣♦st♦♥ π st ♥ ♦r♠ tr♥é sr ΩA(log I).

Pr ♦t x =p∑i=1xiduiui

+n∑

i=p+1xidai ∈ ΩA(log I).

♥

[Φ(x)](x) = [p∑1

xiui[σ H d](ui) +

n∑p+1

xi[σ H d](ai)](x)

=p∑1

xiui[σ H d](ui)[

p∑j=1

xjdujuj

+n∑

j=p+1xjdaj ]+

+n∑p+1

xi[σ H d](ai)[p∑j=1

xjdujuj

+n∑

i,j=p+1xid(aj)]

=p∑

i,j=1

xixjuiuj

σ[H d(ui)] d(uj)+

+n∑

i,j=p+1

xixjui

σ[H d(ui)] d(aj)+

+n∑

i,j=p+1

xixjuj

σ[H d(ai)] d(uj)+

+n∑

i,j=p+1xixj σ[H d(ai)] d(aj)

=p∑

i,j=1

xixjuiuj

ui;uj+n∑

1≤i≤p,p+1≤j≤n

xixjui

ui; aj

+n∑

1≤j≤p,p+1≤i≤n

xixjuj

ai;uj+n∑

i,j=p+1xixjai; aj = 0

t♥t ♦♥♥é q ΩA st ♥ s♦s ♠♦ ΩA(log I), ♠♦ s ♦r♠s

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

tr♥és sr ΩA(log I) st ♦♥t♥ ♥s ♠♦ s ♦r♠s tr♥és sr ΩA.

π ♣t ♦♥ ♦♣érr sr du⊗ dv ♣♦r t♦t u, v ∈ A.♥ ♦♥ π(du, dv) = [Φ(du)]dv = σ(H(du))dv = H(du)v = u, v. ♠♠ ss♦s ♦♥♥ qqs ♣r♦♣rétés π

♠♠ ♦t π ♦r♠ ss♦é à ♥ strtr P♦ss♦♥ ♦rt♠q

♣r♥♣ −,−. P♦r t♦t u, v ∈ S, a, b ∈ A ♦♥

π(adu

u, bdv

v) =

ab

uvu, v

π(adu, bdv

v) =

ab

vu, v

π(adu, bdv) = abu, v

Pr P♦r q st ♣r♠èr ♣r♦♣rété ♦♥sér♦♥s

u, v ∈ S t a, b ∈ A. ♣rès é♥t♦♥ π, ♦♥

π

(adu

u, bdv

v

)= Φ(a

du

u)bdv

v

=ab

uvσ(u,−)dv

=ab

uvu, v

Pr ♥ rs♦♥♥♠♥t ♥♦ ♦♥ é♠♦♥tr s ♣r♦♣rétés rst♥ts

qs ①♠♣s èrs P♦ss♦♥ ♦rt♠qs

♠♠ ♦t (A, −,−0) ♥ èr P♦ss♦♥ P♦r t♦t a0 ∈ A, a0 6=0A,

−,− := a0−,−0 st ♥ strtr P♦ss♦♥ ♥s A s t s♠♥t s ♣♦r

t♦t a, b, c ∈ A,

a, a00b, c0 + b, a00c, a0 + c, a00a, b0 = 0A

Pr t ♠♦♥trr q −,− = a0−,− ér ♥tté ♦ s t

s♠♥t s été st stst Pr rs −,− ér ♥tté

♦ s t s♠♥t s a, b, c+ b, c, a+ c, a, b = 0A. q éqt

à 0 = a, b, c+ b, c, a+ c, a, b

= a0a, a0b, c00 + a0b, a0c, a00 + a0c, a0a, b00= a0(a, a00b, c0 + b, a00c, a0 + c, a00a, b0)+

+a20(a, b, c00 + b, c, a00 + c, a, b00)= a0(a, a00b, c0 + b, a00c, a0 + c, a00a, b0)

réstt s ét ♥térté A.♥ ♣rtr été st t♦♦rs éré ♣♦r t♦t s♠r a0 ∈ A

−,−0.

èrs P♦ss♦♥ ♦rt♠qs

♦r♦r ♦t −,−0 ♥ strtr P♦ss♦♥ ♥s A.a0−,−0 st ♥ strtr P♦ss♦♥ ♦rt♠q ♦♥ é a0A ♣♦r

t♦t a0 ∈ A ér♥t

♥ é♠♥t ♦r♦r s♥t

♦r♦r ♦t A := k[x, y] ♦ A = k[x, y, z] t a0 ∈ A. P♦r t♦t str

tr P♦ss♦♥ −,−0 ♥s A, a0−,−0 st ♥ strtr P♦ss♦♥ ♥s A♦rt♠q ♣r♥♣ ♦♥ a0A.

Pr ❱♦r ♥♥①

①♠♣ r A := C[x, y, t, z] ♦♥ é♥t r♦t

f, g = xyz

(∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x

)+∂f

∂t

∂g

∂z− ∂f

∂z

∂g

∂t

♦♥tr♦♥s q −,− é♥ ♣r st ♥ r♦t P♦ss♦♥ ♥s A.• rè ♥③ t ♥ts②♠étrq é♦♥t été s♥t

xyz

(∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x

)+∂f

∂t

∂g

∂z−∂f∂z

∂g

∂t= xyz

df ∧ dg ∧ dt ∧ dzdx ∧ dy ∧ dt ∧ dz+

df ∧ dg ∧ dx ∧ dydx ∧ dy ∧ dt ∧ dz

• P♦r q st ♥tté ♦ st ♠♦♥trr ♣rès

❬♥r♦③ ❪ q

[π, π] = 0

♦ù [−,−] és♥ r♦t ♦t♥ t π = xyz∂

∂x∧ ∂

∂y+∂

∂t∧ ∂

∂zés♥

tr ss♦é à −,−.P♦r st érr q

[xyz∂

∂x∧ ∂

∂y, xyz

∂

∂x∧ ∂

∂y] = 0, [

∂

∂t∧ ∂

∂z,∂

∂t∧ ∂

∂z] = 0

t [xyz∂

∂x∧ ∂

∂y,∂

∂t∧ ∂

∂z] = 0.

Pr rs ért♦♥ Df := f,− = xyz(∂f

∂x

∂

∂y− ∂f

∂y

∂

∂x) +

∂f

∂t

∂

∂z− ∂f

∂z

∂

∂tér Dz(xyz) = xy /∈ (xyz)A. ♥st ♦♥ ♣s ♥ strtr P♦ss♦♥ ♦

rt♠q ♦♥ (xyz)A. ♣♥♥t st ♦rt♠q ♣r♥♣ ♦♥

(xy)A.

①♠♣ r A := C[x, y, z], ♦♥ s ♦♥♥ ① éé♠♥ts h, p ∈ A ♥♦♥

♦♥st♥ts r♦t

f, ghp := hdf ∧ dg ∧ dpdx ∧ dy ∧ dz +

df ∧ dg ∧ dhdx ∧ dy ∧ dz .

st ♥ér ♥ts②♠étrq t stst rè ♥③

P♦r ♠♦♥trr q r♦t st P♦ss♦♥ st ♦♥ érr ♥tté

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

♦ ♥♥t ♦♠♣t s ♣r♦♣rétés ért♦♥s r♥r st

♠♦♥trr q

z, x, yhphp + x, y, zhphp + y, z, xhphp = 0

P♦r ♦♥ r♠rqr q

x, yhp = h∂p

∂z+∂h

∂z

♥ ♣♦s♥t H = x, yhp, ♥ ♣♣t♦♥ s♠♣ s é♥t♦♥s ♦♥♥

z, x, yhphp = h

(∂H

∂x

∂p

∂y− ∂H

∂y

∂p

∂x

)+

(∂H

∂x

∂h

∂y− ∂H

∂y

∂h

∂x

)

♥ sstt ♥s tt été∂H

∂yt∂H

∂x♣r

∂H

∂x=∂h

∂x

∂p

∂z+ h

∂2p

∂xz+∂2h

∂xz.

∂H

∂y=∂h

∂y

∂p

∂z+ h

∂2p

∂yz+∂2h

∂yz.

t ♦♥ ♦t♥t

z, x, yhphp + x, y, zhphp + y, z, xhphp =

= h2∂p

∂y

∂2p

∂xz+ h

∂p

∂y

∂2h

∂xz− h2

∂p

∂x

∂2p

∂yz− h

∂p

∂x

∂2h

∂yz+∂h

∂y

∂2h

∂xz− h

∂h

∂x

∂p

∂yz− ∂h

∂x

∂h

∂yz+

h2∂p

∂z

∂2p

∂yx+ h

∂p

∂z

∂2h

∂yx− h2

∂p

∂y

∂2p

∂zx− h

∂p

∂y

∂2h

∂zx+∂h

∂z

∂2h

∂yx− h

∂h

∂y

∂p

∂zx− ∂h

∂y

∂h

∂zx+

h2∂p

∂x

∂2p

∂zy+ h

∂p

∂x

∂2h

∂zy− h2

∂p

∂z

∂2p

∂xy− h

∂p

∂z

∂2h

∂xy+∂h

∂x

∂2h

∂zy− h

∂h

∂z

∂p

∂xy− ∂h

∂z

∂2h

∂xy= 0.

♦♥ −,−hp st ♥ ♥ strtr P♦ss♦♥ sr A. ♣s ♣♦r t♦t f ∈ A,♦♥

f, hhp = hdf ∧ dh ∧ dpdx ∧ dy ∧ dz ∈ hA.

♥ ♦♥t ♦♥ q −,−hp st ♦rt♠q ♦♥ hA.

①♠♣ ♦♥t (Qi)1≤i≤n−2 (n − 2) éé♠♥ts C[x1, ..., xn]. P♦r t♦t

λ ∈ C[x1, ..., xn], r♦t

f, g = λdf ∧ dg ∧ dQ1 ∧ ... ∧ dQn−2

dx1 ∧ .... ∧ dxnst ♥ strtr P♦ss♦♥ ♥s

C[x1, ..., xn] ♦rt♠q ♦♥ λC[x1, ..., xn].

Pr♦♣♦st♦♥ ♦t strtr P♦ss♦♥ ♥s C[x, y] st s♦t s②♠♣tq

s♦t ♦rt♠q

Pr ♦t −,− ♥ strtr P♦ss♦♥ ♥s C[x, y].

P♦r t♦t f, g ∈ C[x, y], ♦♥

f, g = x, y(∂f∂x

∂g

∂y− ∂f

∂y

∂g

∂x)

Prt♥t −,− st s②♠♣tq s x, y ∈ C∗. ♥s s ♦♥trr st

♦rt♠q ♦♥ x, yC[x, y].

❱rétés P♦ss♦♥ ♦rt♠qs

❱rétés P♦ss♦♥ ♦rt♠qs

tt ♣rt st ♦♥sré à ♦♥strt♦♥ é♦♠étrq ♥♦t♦♥ strtr

P♦ss♦♥ ♦rt♠q

♥s tt ♣rt s ♠♥t♦♥ ①♣t♦♥♥ ♦♥ és♥r ♣r

• X ♥ rété ♦♠♣① ♠♥s♦♥ ♦♠♣① n,

• OX s s r♠s ♦♥t♦♥ ♦♦♠♦r♣s

• ΩX s s r♠s s ♦r♠s ♦♦♠♦r♣s sr X,

• MD s s r♠s s ♦r♠s ♠ér♦♠♦r♣s sr D.

sr r

♦t U ♥ ♦♠♥ Cn t D ⊂ U ♥ ②♣rsr U é♥ ♣r éq

t♦♥ h(z) = 0, ♦ù h st ♥ ♦♥t♦♥ ♦♦♠♦r♣ ♦rs ♣♦r t♦t q♦r♠ ω

♠ér♦♠♦r♣ ♥s U à ♣ôs ♥s D, ♦♥ té♦rè♠ s♥t

é♦rè♠ ❬t♦ ❪ s ♣r♦♣rétés s♥ts s♦♥t éq♥ts

hω t hdω s♦♥t ♦♦♠♦r♣s

hω t dh ∧ ω s♦♥t ♦♦♠♦r♣s

①st ♥ ♦♥t♦♥ ♦♦♠♦r♣ g t ♥ (q − 1)♦r♠ ξ t ♥ q♦r♠

♦♦♠♦r♣ η sr U t q

dimCD ∩ z ∈ U : g(z) = 0 ≤ n− 2

gω =dh

h∧ ξ + η

①st ♥ s♦s s♣ ♥②tq ♠♥s♦♥ (n − 2) A ⊂ D t q s

r♠s ω ♥ t♦t ♣♦♥t p ∈ D−A s♦♥t ♦♥t♥s ♥sdh

h∧Ωq−1

U,p +ΩqU,p.

té♦rè♠ st é♥t♦♥ s♥t

é♥t♦♥ ❯♥ q♦r♠ ♠ér♦♠♦r♣ sr U st ♦rt♠q ♦♥ D s

stst s ♦♥t♦♥s éq♥ts é♦rè♠

P♦r t♦t ♣♦♥t p X t t♦t ♥tr ♥tr q, ♦♥ ♥♦t

ΩqX,p(logD) := r♠ s q♦r♠s ♦rt♠qs ♥ ♣

ΩqX(logD) := ∪p∈X

ΩqX,p(logD)

Pr rs ♣r♦♣♦st♦♥ s♥t ♥♦s ♣r♠t é♥r ♥♦ é♦♠étrq

♥♦t♦♥ ért♦♥ ♦rt♠q ♥tr♦t à ♣rt

Pr♦♣♦st♦♥ ❬t♦ ❪ ♦t δ ♥ ♠♣s trs sr X s ♣r♦♣rétés

s♥ts s♦♥t éq♥ts

P♦r t♦t ♣♦♥t ss p D, tr t♥♥t δ(p) ♥ p st t♥♥t à D.

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

P♦r t♦t ♣♦♥t p D, s hp st ♦♥t♦♥ é♥t♦♥ D, ♦rs δhp st

♥s é (hp)OX,p.

é♥t♦♥ ❯♥ ♠♣s trs δ st t ♦rt♠q ♦♥ D ♦ ♦

rt♠q s ér s ♦♥t♦♥s éq♥ts Pr♦♣♦st♦♥

♥ ♣♦s

DerX,p(logD) ④δ r♠ s ♠♣s trs ♦♦♠♦r♣s sr X ♥ p t q

δhp ∈ hpOX,P ⑥

DerX(logD) = ∪p∈X

DerX,p(logD)

♠♠ ♥ s ♣r♦♣rétés s♥ts

DerX(logD) st ♥ OXs♦s ♠♦ ♦ér♥t DerX .

DerX(logD) st st ♣♦r r♦t [−,−] s ♠♣s trs

♦♦♠♦r♣s

Pr ♣r♠èr ♣r♦♣rété é♦ t q DerX(logD) st ♥♦②

♠♦r♣s♠ s① ♦ér♥ts s♥t

DerX → OX/hOX

δ 7→ δh.

①è♠ ♣r♦♣rété st rt

♠♠ s♥t étt ♥ ♥ ♥tr s s ♦r♠s ér♥ts ♦rt

♠qs t s ♠♣s trs ♦rt♠qs

♠♠ ❬t♦ ❪

éré ♥ ♦r♠ ♦rt♠q s♥t ♥ ♠♣ tr

♦rt♠q st ♥ ♦r♠ ♦rt♠q

♦♥trt♦♥ ♥ ♦r♠ ♦rt♠q ♣r ♥ ♠♣ trs ♦rt

♠q st ♥ ♦r♠ ♦rt♠q

♥ ♣rtr ♦♥trt♦♥ ♥t ♥ té ♥tr DerX,p(logD) t

Ω1X,p(logD) ♣♦r t♦t p ∈ D.

s♥st ss q ΩX,p(logD) t DerX,p(logD) s♦♥t s OX,p♠♦s ré①s

♥ é♥ér ΩX,p(logD) tDerX,p(logD) ♥ s♦♥t rs q sD ér s ②♣♦tèss

té♦rè♠ ♥s ❬t♦ ❪

é♥t♦♥ ❯♥ sr rét D X st t r ♦ t♦ s DerX,p(logD)

st r ♥ t♦t p ∈ D.

①♠♣ ♥ ♦♥sèr sr X = C3 sr D = h = 0 ♦ù h = xy(x+

y)(y+xz) s ♠♣s trs δ1 = x∂x+y∂y, δ2 = x2∂x−y2∂y−z(x+y)∂zt δ3 = (xz + y)∂z ér♥t δ1(h) = 4f, δ2(h) = (2x − 3y)h t δ3(h) = xh. Pr

rs δ1 ∧ δ2 ∧ δ3 = −xy(zx+ y)(y + x). ♥ ♦♥t q D st ♥ sr r

X.

❱rétés P♦ss♦♥ ♦rt♠qs

♣r♦♣♦st♦♥ s♥t ét ♥ ♥ ♥tr s ért♦♥s ♦rt♠qs ♣r♥

♣s t s ♠♣s trs ♦rt♠qs

Pr♦♣♦st♦♥ ♦t D ♥ sr X. ♦t ♠♣s tr ♦rt♠q

♦♥ D st ♥ ért♦♥ ♦rt♠q ♣r♥♣ OX .

Pr st r q t♦t ♠♣ trs sr X st ♥ ért♦♥ OX .

♦t δ ♥ ♠♣ trs ♦rt♠q ♦♥ D. ♥ s♣♣♦s q D :=

z;h(z) = 0 t q S = h1, ..., hp ♦ù h = h1.h2...hp ♣rès é♥t♦♥

δ(hi) ∈ hiOX . ♦♥ δ st ♦rt♠q ♣r♥♣ ♦♥ S. é♦rè♠ ♥♦s és♦♥s q t♦t ♦r♠ ♦rt♠q ω ♠t ♥

értr ♦r♠

gω =dh

h∧ ξ + η.

♥ ♥ ét é♥t♦♥ s♥t

é♥t♦♥ rés ♥ q♦r♠ ♦rt♠q ω st rstrt♦♥ ξ

gà D.

♥ ♥♦tr resω

é♦rè♠ s♥t rtérs s srs ♣♦r sqs ΩX(logD) st ♥♥ré

♣r s ♦r♠s r♠és

é♦rè♠ ❬t♦ ❪ ♦t (D, p) = (D1, p)∪ ...∪ (Dm, p) é♦♠♣♦st♦♥

♦ ♥ ♦♠♣♦s♥ts rréts ♥ sr D ♥ ♥ ♣♦♥t p ∈ D, t h = h1...hm s ♦♥t♦♥ é♥t♦♥

s ♦♥t♦♥s s♥ts s♦♥t éq♥ts

Ω1X,p(logD) =

m∑i=1

OX,pdhihi

+Ω1X,p

Ω1X,p(logD) st ♥♥ré ♣r s ♦r♠s r♠és

res(Ω1X,p(logD)) =

n⊕i=1

ODi,p

Di st ♥♦r♠ dimCSingDi ≤ n− 3 ♣♦r ♠

Di ⋔ Dj i 6= j; i, j = 1, ...m sr ♦♠♣é♠♥tr ♥ s♦s

♥s♠ ♠♥s♦♥ n− 3 D Di t Dj s♦♥t à r♦s♠♥ts ♥♦r♠①

♣♦r i 6= j, i, j = 1, ...,m

dimCDi ∩Dj ∩Dk ≤ n− 3 ♣♦r i 6= j 6= k 6= i i, j, k = 1, ...,m.

❯♥ r♠rq sr s ♦r♠s ér♥ts ♦rt♠qs t♠♣s trs ♦rt♠qs

♥s tt s♦s st♦♥ ♥♦s ♣♣♦rt♦♥s qqs ♣rés♦♥s sr ♥♦t♦♥

♦r♠s ér♥ts ♦rt♠qs

t♥t ♦♥♥é q ω =dy

xér

x2ω = xdy ∈ ΩX t dx2∧ω = 2xdx∧ dyx

= 2dx∧dy ∈ ΩX ♦♥ ♣t ♦♥r q

st ♦rt♠q ♦♥ sr D C2 é♥ ♣r ♦♥t♦♥ ♦♦♠♦r♣

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

h(x, y) = x2.

r éqt♦♥ gω = 2adx

x+bdx+cdy ♥ s ♦♥t♦♥s ♦♦♠♦r♣s g, a, b t c ♣♦r

s♦t♦♥

g = xc

2a+ xb = 0

♥t ♦♥ q ♠♥s♦♥ D ∩ (x, y) ∈ C2, g(x, y) = 0 st 1; ♣♦r t♦t

s♦t♦♥ (g, a, b, c) 6= (0, a, b, 0).

s♥st ♣rès ♣r♦♣rété é♦rè♠ q ♣♦r t♦ts ♦♥t♦♥s

♦♦♠♦r♣s g, a t t♦t ♦r♠ ♦♦♠♦r♣ η ts q gω = 2adx

x+ η, ♦♥

1 = dimC(D ∩ (x, y) ∈ C2, g(x, y) = 0) ≤ 2 − 2 = 0. q st sr ♦♥

ω =dy

x♥st ♣s ♥ ♦r♠ ♦rt♠q ♦♥ D ♦rsq ♦♥ s♣♣♦s q

D st é♥ ♣r ♦♥t♦♥ h = x2. ♦♥trt éq♥ s ♣r♦♣rétés

é♦rè♠ tt ♦♥trt♦♥ rést t q ♦♥t♦♥ é♥t♦♥

D ♥st ♣s rét s♥st q t ♦tr ♦♥t♦♥ rrétté D

♥s s ②♣♦tèss té♦rè♠ ❬t♦ ❪ ♥s t♦t q st ♥♦s

s♣♣♦sr♦♥s q ♦♥t♦♥ é♥t♦♥ D st à rré r éqt♦♥ ♦

hp = 0 ♥ t♦t ♣♦♥t p ∈ D.

é♥t♦♥ t ♣r♠èrs ♣r♦♣rétés

♦t X ♥ rété ♦♠♣① ♠♥s♦♥ ♥ n t D ♥ sr rét t r

X éqt♦♥ h = 0 ♦ù h st r♠ ♥ ♦♥t♦♥ ♦♦♠♦r♣ ♥ ♥♦t OX

s s r♠s ♦♥t♦♥s ♦♦♠♦r♣s sr X.

é♥t♦♥ ❯♥ strtr P♦ss♦♥ ♦♦♠♦r♣ sr X st ♦♥♥é ♥ r♦

t −,− q ss♥ à ♥ ♦♣ (f, g) r♠s ♦♥t♦♥s ♦♦♠♦r♣s ♥ ♥

♣♦♥t x X ♥ r♠ f, g ♦♥t♦♥ ♦♦♠♦r♣ ♥ x ér♥t s ♣r♦♣rétés

s♥ts

• −,− st ♥ér ♥ts②♠étrq

• f, g, h+ g, h, f+ h, f, g = 0 ♥tté ♦

• f, gh = f, gh+ f, hg rè ♥③

st ♣r♦é ♥s ❬P♦s ❪ q t♦t strtr P♦ss♦♥ ♦♦♠♦r♣

♥t ♥ ♦♠♦♠♦r♣s♠ OX ♥ér

H : ΩX → DerX

t q H(df)(g) = f, g H st ♣♣t♦♥ ♠t♦♥♥♥ ss♦é à −,− tt ♣♣t♦♥ ♦♥ ♠♦♥tr q t♦t strtr P♦ss♦♥ ♦♦♠♦r♣

♥t ♥ t♥sr ♦♦♠♦r♣

π ∈ H0(X,2∧TX)

♣♣é tr P♦ss♦♥

❱rétés P♦ss♦♥ ♦rt♠qs

é♥t♦♥ ❯♥ strtr P♦ss♦♥ ♦♦♠♦r♣ −,− sr X st t ♦

rt♠q ♦♥ D s ♣♦r t♦t r♠ f ♦♥t♦♥ ♦♦♠♦r♣ ♠♣

♠t♦♥♥ ss♦é H(df) st ♥ st♦♥ DerX(logD)

♥s st t♦t rété P♦ss♦♥ ♦♦♠♦r♣ ♦rt♠q ♦♥ ♥

sr D sr ♣♣é s♠♣♠♥t rété P♦ss♦♥ ♦rt♠q t ♦♥ ♥♦tr

(X, −,−, D).

é♦ tt é♥t♦♥ q ♣♦r t♦t ♦rt U X t t♦t st♦♥ f

OX sr U f,− st ♥ ért♦♥ ♦rt♠q ♣r♥♣ ♦♥ é

é♥t♦♥ D.

Psq D st r é♦rè♠ ♥s ❬t♦ ❪ ♥tr♥ q

n∧ Ω1

X(logD) = ΩnX(logD).

t

DeriX(logD) :=i∧Der1X(logD).

♥s s ♦♥

ΩqX(logD) =q∧Ω1

X(logD) ∼= HomOX(q∧Der1X(logD),OX)

é♥t♦♥ ♦t D ♥ sr r X.

s st♦♥s q∧Der1X(logD) s♦♥t ♣♣és q♠♣s trs ♦rt♠qs

♥ ♣♦s erX(logD) :=n⊕i=1DeriX(logD)

[−,−]s és♥ r♦t ♦t♥ ♦rs ♦♠♣t t♥ t qDerX(logD)

st st ♣♦r r♦t s ♠♣s trs erX(logD) rst st

♣♦r [−,−]s.

é♥t♦♥ ♥ ♣♣ r♦t ♦t♥ ♥s ♦rt♠q ♦♥ ♥

sr r D rstrt♦♥ [−,−]s à erX(logD).

s♥st q♥ tr ♦♦♠♦r♣ ♦rt♠q π st P♦ss♦♥ s t s♠♥t

s s♦♥t r♦t ♦t♥ ♦rt♠q st ♥

♦r♦r tr P♦ss♦♥ t♦t strtr P♦ss♦♥ ♦rt

♠q sr X st ♥ st♦♥ Der2X(logD)

Pr ♦t π tr ♥ strtr P♦ss♦♥ ♦rt♠q sr X, ♦rs

♣♦r t♦t a, b ∈ OX , ♦♥

π(da, db) := H(da)b

stàr idaπ ∈ Der1X(logD).

♥ ét ♣r♦♣rété ♥rs ♦♣ (ΩX , d) q DerX ∼=σHom(ΩX ,OX).

♦♠♣t t♥ t q ΩX ⊂ ΩX(logD) ♦rs Hom(ΩX(logD),OX) ⊂Hom(ΩX ,OX). Pr rs ♥s♦♥ Der1X(logD) ♥s DerX ♠♣q q

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

Hom(ΩX(logD),OX) ∼= Der1X(logD) ∼= σ(Der1X(logD)). ♥s H ♥t ♥

♦♠♦♠♦r♣s♠ ΩX rs Hom(ΩX(logD),OX); r♥r s ♣r♦♦♥ ♦♥

♠♥èr ♥♦♥q ♥ ♥ ♦♠♦♠♦r♣s♠ s① OX ♠♦s H

ΩX(logD) rs Hom(ΩX(logD),OX). ♦ù réstt

♦r♦r ♦t strtr P♦ss♦♥ ♦♦♠♦r♣ ♥♦♥ tr t ♥♦♥ s②♠

♣tq sr ♥ sr ss st ♦rt♠q

Pr

♦s s♦♥s q t♦t strtr P♦ss♦♥ ♦♦♠♦r♣ ♥♦♥ ♥ sr X st ♥t

♣r ♥ st♦♥ π ré ♥t♥♦♥q ω−1X X. Psq π st ♥♦♥ s②♠♣tq

①st ♥ ♦♥t♦♥ ♦♦♠♦r♣ h t qD := z ∈ X,h(z) = 0 t π = h∂x∧∂y. s♥st q π st ♦rt♠q ♦♥ D.

①♠♣ ♦t D = h = x4 + y5 + xy4 = 0 ♥ ♦r ♣tq

X = C2. s ♠♣s trs δ1 = (16x2 + 20xy)∂x + (12xy + 16y2)∂y t

δ2 = (16xy2 + 4y3 − 12xy)∂x + (12y3 − 4x2 + 5xy − 100y2)∂y s♦♥t ♦rt♠qs

♦♥ D s s♦♥t rs t ♦♥stt♥t ♦♥ ♥ s DerX(logD). q

♠♣q q D st r ♥ é♥t sr C2 r♦t P♦ss♦♥ s♥t

f, g = −(64x4 + 1356x2y2 + 64xy4 + 1808xy3 + 64y5)(∂xf∂yg − ∂yf∂xg).

r♦t st P♦ss♦♥ ♦rt♠q ♦♥ D.

♥ t

f, g =

= [(16x2 + 20xy)(12y3 − 4x2 + 5xy − 100y2)− (16xy2 + 4y3 − 12xy)(12xy + 16y2)]

(∂xf∂yg − ∂yf∂xg)

=

∣∣∣∣16x2 + 20xy 12xy + 16y2

16xy2 + 4y3 − 12xy 12y3 − 4x2 + 5xy − 100y2

∣∣∣∣ (∂xf∂yg − ∂yf∂xg)

= hk(∂xf∂yg − ∂yf∂xg)

♦ù

hk =

∣∣∣∣16x2 + 20xy 12xy + 16y2

16xy2 + 4y3 − 12xy 12y3 − 4x2 + 5xy − 100y2

∣∣∣∣ .

①st♥ k st ssré ♣r t q D st r

s♥st q ♣♦r t♦t ♦♥t♦♥ ♦♦♠♦r♣ f, f,− = kh(∂xf∂y − ∂yf∂x) r

kh(∂xf∂y − ∂yf∂x) ∈ DerX(logD). ♥ ♦♥t q tt strtr P♦ss♦♥ st

♦rt♠q ♦♥ D.

①♠♣ ♥ ♦♥sèr sr X = C3 r♦t f, g = (zx+y)(x(∂xf∂zg−∂zf∂xg)−y((∂yf∂zg−∂zf∂yg)). ♦♥tr♦♥s q st P♦ss♦♥ ♦rt♠q ♦♥

sr D = h = xy(x+ y)(y + xz) = 0 X = C3.

t♥sr ss♦é à r♦t st

π = x(zx+ y)∂x∧ ∂y+ y(xz+ y)∂y ∧ ∂z. P♦r ♠♦♥trr q −,− st P♦ss♦♥

st ♠♦♥trr q

πhi∂hπjk+πhj∂hπki+πhk∂hπij = 0 ♣♦r t♦s i, j, k = 1, 2, 3 ♦ù (πij) st ♠tr

π. ♥s s ♣rtr s étés s♦♥t éq♥ts à

❱rétés P♦ss♦♥ ♦rt♠qs

z, x∂zy, z+ z, y∂zz, x = 0. q st éré

♦s ♣♦♦♥s ss r♠rqr q D st r r s ♠♣s trs δ1 =

x∂x + y∂y δ2 = x2∂x − y2∂y − z(x + y)∂z t δ3 = (xz + y)∂z ♦r♠♥t ♥ s

DerX(logD) t q π = δ1 ∧ δ3. ♥ ♣t ♦♥ r r♦t ♦t♥

♦rt♠q π. r r♦t ♥♦s ♦♥♥ [δ1 ∧ δ3, δ1 ∧ δ3] = [δ1, δ1] ∧ δ3 ∧δ3 + δ ∧ [δ1, δ3] ∧ δ3 + δ ∧ [δ3, δ1] ∧ δ3 + δ1 ∧ δ1 ∧ [δ3, δ3] = 0. q ♠♦♥tr q

r♦t st P♦ss♦♥ rst à ♠♦♥trr q st ♦rt♠q ♦♥ D.

P♦r st r♠rqr q ♣♦r t♦t ♦♥t♦♥ ♦♦♠♦r♣ sr X, ♦♥

f,− = δ1(f)δ3 − δ3(f)δ1 q st ♦rt♠q ♦♥ D.

❱rétés ♦s②♠♣tqs

♥s tt ♣rt D és♥r ♥ sr r ♥ rété ♦♠♣① X

♠♥s♦♥ ♦♠♣① n t ω és♥r ♥ ♦r♠ ♦rt♠q ♦♥ D

r♠é ♥ ♦♥sèr ♠♦r♣s♠ s① I : DerX(logD) −→ ΩX(logD)

é♥ ♣r

I(v) = ivω.

P♦r t♦t v ∈ DerX(logD), ♦♥ ♥♦t Lvω éré ω s♥t v.

é♥t♦♥ ❯♥ st♦♥ v DerX(logD) st t ω♦s②♠♣tq s

♣résr ω Lvω = 0.

♥s♠ s ♠♣s ω♦s②♠♣tqs sr ♥♦té SympωX ♣r♦♣♦st♦♥ s

♥t ♦♥♥ ♥ rtérst♦♥ s r♠s ♠♣s ω♦s②♠♣tqs

Pr♦♣♦st♦♥ ❯♥ ♠♣ trs ♦rt♠q v st ω

♦s②♠♣tq s t s♠♥t s ivω st ♥ ♥ ♦r♠ ♦rt♠q r

♠é

♦t α r♠ ♦r♠ ♦rt♠q sr X. ①st v ∈ SympωX t

q α = I(v) ♦rs iwα = 0 ♣♦r t♦t w ∈ ker(I).

Pr t♥t ♦♥♥é q dω = 0 ♦rs

Lv(ω) = ivdω + divω = d(I(v))

è ♣r ♣r♠èr ♣r♦♣rété

P♦r q st ①è♠ ♣r♦♣rété ♦♥ ♣♦r t♦t w ∈ ker(I)

iwα = α(w)

= I(v)(w)

= ω(v, w)

= −ω(w, v)= −I(w)(v) = 0

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

é♥t♦♥ ❯♥ ♠♣ trs ♦rt♠q v st t ω♠t♦♥♥ s ①

st ♥ ♦♥t♦♥ ♦♦♠♦r♣ f sr X t q I(v) = df

❯♥ t ♦♥t♦♥ ♦rsq ①st st ♣♣é ω♠t♦♥♥♥ v.

♣r Pr♦♣♦st♦♥ ♥♦s és♦♥s q s ♠♣s ω♠t♦♥♥s

s♦♥t ωs②♠♣tqs

és♥♦♥s ♣rHωX ♥s♠ s ♠♣s ω♠t♦♥♥s t ♣rH1(X, logD) ♣r

♠r r♦♣ ♦♦♠♦♦q ♠ ♦rt♠q X. ♥ ♣r♦♣♦st♦♥

s♥t

Pr♦♣♦st♦♥ st 0 → HωX → SympωX → H1(X, logD) st

①t

♦rsq D st ♦♠♥t qs♦♠♦è♥ t (X−D) ♣r♦♠♣t tt st

♥t

0 → HωX → SympωX → H1(X −D,C)

Pr ♣r♠èr ♣r♦♣rété é♦ t q i[v,w]ω = d(iv(dg)) ♣♦r t♦t

v, w ∈ HωX ts q I(v) = df t I(w) = dg. ①è♠ ♣r♦♣rété q♥ à

é♦ é♦rè♠ r♦t♥ ♠ t é♦rè♠ ♦♠♣rs♦♥

♦rt♠q

①♠♣ P♦r D := (0, z2, z3) ∈ C3, ♦♥

v = v1z1∂z1 + v2∂z2 + v3∂z3 ∈ DerX(logD).

Pr rs ω =dz1z1

∧ dz2 +dz1z1

∧ dz3 ∈ ΩC3(logD) dω = 0 t I(v) = −(v2 +

v3)dz1z1

+ v1d(z2 + z3). ♥ ♣r♥♥t v1 = 0 t v2 + v3 = −1 v rst ♥ ♠♣

trs ♦rt♠q ♦♥ D t ♦♥ I(v) =dz1z1

= d log z1 Psq ♦♥t♦♥

(z1, z2, z3) 7→ log z1 ♥st ♣s ♦♦♠♦r♣ sr C3, ♦rs v = v2∂z2 − (1 + v2)∂z3 ♥st

♣s ♥ ♠♣ ω♠t♦♥♥

♥ ♣♦s K = ker(I).

ω st r♥ ♦♥st♥t t ♥♦♥ tr s ♦♥t♦♥s ω♠t♦♥♥♥s ①st♥t

♦♠♥t ♥ ♣t ♦♥ ♥tr♦r s s r♠s ♦♥t♦♥s ♦♠♥t

ω♠t♦♥♥♥s

és♥♦♥s ♣r OX/K s♣ s ♦♥t♦♥s ♦♠♥t ω♠t♦♥♥♥s

Pr♦♣♦st♦♥ OX/K st ♥ s èrs P♦ss♦♥

Pr ♦♥t f, g ∈ OX/K ; ①st v, w ∈ DerX(logD) ts q df = I(v) t

dg = I(w) r d(fg) = fdg + gdf = fI(w) + gI(v) = I(fw + gv). ♦♥ OX/K st

♥ s♦s èr OX . ♣rès é♥t♦♥ OX/K , ♣♣t♦♥ ϕ : v 7→ f ♦ù

df = I(v) st ♥ srt♦♥ OX/K sr HωX

❱rétés P♦ss♦♥ ♦rt♠qs

①st ♥ ♣♣t♦♥ ψ : OX/K → HωX t q ϕ ψ = idOX/K

♥ ♦♥sèr ♣♣t♦♥ ♥ér

−,−ω : OX/K ⊗OX/K → OX/K

(f, g) 7→ ψ(f)g

♣rès q ♣réè ♦♥

f, gω = ψ(f).g

= ω(w,ψ(f))

= −ω(ψ(f), w)= −iwiψ(f)ω = −iψ(g)df = −g, fω

P♦r q st ♥tté ♦ ♥♦s ♦♥s

(dω)(ψ(f), ψ(g), ψ(h)) = ψ(f)ω(ψ(g), ψ(h)) − ψ(g)ω(ψ(f), ψ(h)) +

ψ(h)ω(ψ(f), ψ(g)) − ω([ψ(f), ψ(g)], ψ(h)) + ω([ψ(f), ψ(h)], ψ(g)) −ω([ψ(g), ψ(h)], ψ(f))

r−ω([ψ(f), ψ(g)], ψ(h)) + ω([ψ(f), ψ(h)], ψ(g))− ω([ψ(g), ψ(h)], ψ(f))

= i[ψ(f),ψ(g)]ωψ(h) + i[ψ(f),ψ(h)]ωψ(g)− i[ψ(g),ψ(h)]ωψ(f)

= −d(iψ(f)dg)ψ(h) + d(iψ(f)dh)ψ(g)− d(iψ(g)dh)ψ(f)

= −d(ψ(f)dg)ψ(h) + d(iψ(f)dh)ψ(g)− d(iψ(g)dh)ψ(f)

= −d(ω(ψ(g), ψ(f)))ψ(h) + d(ω(ψ(h), ψ(f)))ψ(g)− d(ω(ψ(h), ψ(g)))ψ(f)

= −ψ(h)ω(ψ(g), ψ(f)) + ψ(g)ω(ψ(h), ψ(f))− ψ(f)ω(ψ(h), ψ(g))

= ψ(f)ω(ψ(g), ψ(h))− ψ(g)ω(ψ(f), ψ(h)) + ψ(h)ω(ψ(f), ψ(g)) .♦♥

(dω)(ψ(f), ψ(g), ψ(h)) = 2(ψ(f)ω(ψ(g), ψ(h)) − ψ(g)ω(ψ(f), ψ(h)) +

ψ(h)ω(ψ(f), ψ(g))).

♥ ♥ ét qf, gω, hω+ = −h, ψ(f)g+ = −ψ(h)(ψ(f)g)+

= −ψ(h)(ψ(f)g)− ψ(f)(ψ(g)h)− ψ(g)(ψ(h)f)

= −ψ(h)ω(ψ(g), ψ(f))− ψ(f)ω(ψ(h), ψ(g))− ψ(g)ω(ψ(f), ψ(h))

= ψ(f)ω(ψ(g), ψ(h))− ψ(g)ω(ψ(f), ψ(h)) + ψ(h)ω(ψ(f), ψ(g))

=1

2(dω)(ψ(f), ψ(g), ψ(h))

= 0r ω st r♠é

♥ ♥ ét ♦r♦r s♥t

♦r♦r (X, −,−ω, D) st ♥ rété P♦ss♦♥ ♦rt♠q

és♥♦♥s ♣r Kω s♦s s èrs K ♦r♠é s ♠♣s ♦①

♦rs ♦♥ st ①t ♦rt s① èrs s♥t

0 → Kω → HωX →

OX/K

C→ 0

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

é♥t♦♥ ♥ ♣♣ rété ♦s②♠♣tq t♦t tr♣t (X,ω,D) ♦r♠é ♥

rété ♦♠♣① ♠♥s♦♥ ♦♠♣① 2n ♥ sr rét D X t ♥

♦r♠ ♦rt♠q r♠é ω ér♥t

ωn 6= 0 ♥s H0(X,Ω2n[D])

♦rsq (X,ω,D) st ♥ rété ♦s②♠♣tq st ♥t

0 → CX → OX → HωX → 0

Pr rs t♦t st♦♥ s : HωX → OX ①t♥s♦♥ ♥t ♥ ♦r♠

C : ∧2HωX → C

é♥ ♣r

C(v, w) = [s(v), s(w)]− s([v, w])

q st ♥ ♦② ②♥r ♦s ♦♥s ♠♦♥trr q st ♥s

ss ♦♦♠♦♦ ω.

Pr♦♣♦st♦♥ C t ω ♦♥t ♠ê♠ ss ♦♦♠♦♦

Pr ♣♣♦♥s q st qst♦♥ tr♦r ♥ ♥ ♥tr ♦r♠ C sr HωX

♥t ♣r t♦t st♦♥ ♥ér s ①t♥s♦♥ èr HωX s ♠♣s

trs ♦♠♥t ♦♠t♦♥♥s t ♦r♠ ♦s②♠♣tq ω sr rété

♦s②♠♣tq X.

Ps ♣résé♠♥t ①t♥s♦♥ st ♦♥♥é ♣r

0 → CXi→ OX

χ→ HωX → 0

♦ù iχfω = df ♣♦r t♦t f ∈ OX .

♣♣♦♥s ss q strtr P♦ss♦♥ ♥t ♣r ω st é♥ ♣r

f, g = −ω(χf , χg).

♣♣t♦♥

θ : HωX → End(OX)

X 7→ θ(X) : f 7→ s(X), f

♥t sr OX ♥ strtr HωX ♠♦

♥ t ♣♦r t♦t X,Y ∈ HωX , ♦♥

θ([X,Y ])f

= s(X), s(Y ), fJacobi= s(X), s(Y ), f − s(Y ), s(X), f= [θ(X), θ(Y )]f.

❱rétés P♦ss♦♥ ♦rt♠qs

♣rés♥t ♣♦s♦♥s Lalt∗(HωX ,OX) ♥s♠ s ♣♣t♦♥s ♠t♥érs

tr♥és sr HωX rs ♥s OX .

Lalt∗(HωX ,OX) ♠♥ ér♥t ②♥r δ é♥ ♣r

δf(X1, ..., Xp) =∑

(−1)i+1θ(Xi)f(X1, ..., Xi, ..., Xp)+∑(−1)i+jf([Xi, Xj ], X1, ..., Xi, ..., Xj , ..., Xp)

st ♥ ♦♠♣① ♥s ♦♥t s r♦♣s ♦♦♠♦♦s ss♦és s♦♥t ♥♦tés

H∗(HωX ,OX).

P♦r p = 1, 2, été ♦♥♥

δf1(X1, X2) = θ(X1)f1(X2)− θ(X2)f

1(X1)− f1([X1, X2])

♣♦r t♦t f1 ∈ Lalt1(HωX ,OX) t

δf2(X1, X2, X3) =

= θ(X1)f2(X2, X3)− θ(X2)f

2(X1, X3) + θ(X3)f2(X1, X2)−

f2([X1, X2], X3) + f2([X1, X3], X2)− f2([X2, X3], X1)

♣♦r t♦t f2 ∈ Lalt2(HωX ,OX).

s st♦♥s ♥érs ①t♥s♦♥ ét♥t s ♣♣t♦♥s C♥érs HωX

rs OX , s s♦♥t ♦♥ s ♦♥s ♦t s ♥ st♦♥ ♣rès

♦♥

δs(X1, X2) = θ(X1)s(X2)− θ(X2)s(X1)− s([X1, X2])

= fX1 , s(X2) − fX2 , s(X1)s([X1, X2])

= −ω(χ(fX1), χ(s(X2))) + ω(χ(fX2), χ(s(X1)))− s([X1, X2])

.

♥s

δs(X1, X2) + ω(X1, X2) = −ω(X1, X2)− s([X1, X2]).

r

C(X1, X2) = s(X1), s(X2) − s([X1, X2]) = −ω(X1, X2)− s([X1, X2])

s étés t ♦♥ ét q

C = ω + δs.

Pr rs ♥ r♠♣ç♥t f2 ♥s ♣r ω t ♥ ♣♣q♥t t q χ st

♥ ♠♦r♣s♠ èrs ♦♥ ♦t♥t

δω = 0.

♦ù réstt

♦t ω =dh

h∧ ψ + η ♥ ♦r♠ ♦s②♠♣tq sr X.

♥ ♣♦s

SD = δ ∈ DerX(logD), ψ.δ = 0.♥

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

♠♠ P♦r t♦t rété ♦s②♠♣tq (X,D, ω), SD st ♥ s♦s

èr DerX(logD)

Pr ♦t ω =dh

h∧ψ+η ♥ ♦r♠ ♦s②♠♣tq sr X. ♥ 0 = dω = dψ.

r

0 = dψ(x, y) = X.ψ(ω)(Y )−X.ψ(Y )− ψ([X,Y ]).

♦ù réstt

és♥♦♥s ♣r Dsing ♣rt s♥èr D t ♣r Dred s ♣rt ss

♥

♦r♦r SD st ♥ strt♦♥ ♥tér X à s ♠♥s♦♥

♥ sr Dred.

qs ①♠♣s rétés P♦ss♦♥ ♦rt♠qs

♥ s♣♣♦s q X st ♥ rété ♦♠♣① D ♥ sr rét t r

X. ♦s tt ②♣♦tès ΩX(logD) rs♣ DerX(logD) ♣t êtr ♦♠♠

s s st♦♥s ♥ ré t♦r T ∗(logD) rs♣ T (logD) T ∗(logD) rs♣

T (logD) st ♣♣é ré ♦t♥♥t t♥♥t ♦rt♠q X. ♥ s ♦♥♥

θ ∈ H0(X,∧2 T (logD)).

Pr é♥t♦♥ θ st ♥ ♣♣t♦♥ OX ♥ér ♥ts②♠étrq sr T ∗(logD).

♦s s♦♥s q ♣♦r t♦t A♠♦M s ♦♥trsM⊗− t (−)M s♦♥t ♦♥ts

♥ tr ♣♦r t♦t R♥♥ A ♥ ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥

HomOX(T ∗(logD))⊗ T ∗(logD)),OX) ≃ HomOX

(T ∗(logD)), T (logD))

Pr ♦♥sérr s♦♠♦r♣s♠ ♦♥t♦♥ ♦♥t♦r t tsr té ♥

tr T ∗(logD) t T (logD)

Pr♦♣♦st♦♥ ♥t q s ♦♥♥r ♥ tr ♦rt♠q π st

éq♥t à s ♦♥♥r ♥ ♥q ♠♦r♣s♠ π : T ∗X(logD) → TX(logD) r♥♥t

♦♠♠tt r♠♠ s♥t

OT ∗X(logD)X ⊗ T ∗

X(logD)ev

T∗X (logD)

OX // OX

T ∗X(logD)⊗ T ∗

X(logD)

π×idT∗X

(logD)

OO

π

77n

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

n

♥ t♥♥t ♦♠♣t é♥t♦♥ evT ∗X(logD)

OX(π) ♦♠♠ttté

r♠♠ ♠♦♥tr q π st é♥ ♣r éqt♦♥ s♥t

〈π(α), β〉 = 〈π, α ∧ β〉

❱rétés P♦ss♦♥ ♦rt♠qs

P♦r t♦t α, β ∈ T ∗X(logD).

Pr♦♣♦st♦♥ ♦t X ♥ rété ♦♠♣① t D ♥ sr rét t r

X ré ♦t♥♥t ♦rt♠q T ∗X(logD) st ♦s②♠♣tq sr π∗(D) st

♦♥ ♥ rété P♦ss♦♥ ♦rt♠q

Pr ♥ t s F : (X1, D1) → (X2, D2) st ♥ ♠♦r♣s♠ rétés ♦♠

♣①s t q F ∗(D2) = D1 ♦rs r♠♠ ♣r♦t ré s♥t ♥t ♥

♦♠♦♠♦r♣s♠

ϕF : X1 ×X2 T∗X logD2 → T ∗

X logD1

T ∗ logD1

X1 ×X2 T∗ logD2

ϕF=F ∗p244

p2 //

p1

T ∗ logD2

π2

F ∗

OO

X1 F// X2

.

♥ ♣♦s♥t ♥s r♠♠ X1 = T ∗(logD) t X2 = X ♦♥ ♦t♥t ♣r s

♣♣t♦♥ ♦♥ ∆ : T ∗(logD) → T ∗(logD) ×X T ∗(logD) r♠♠

♦♥♥♥t à ♥ ♠♦r♣s♠ s① θ = ϕπ ∆

T ∗X(logD)

θ //

∆

T ∗T ∗X(logD)(log π

∗(D))

T ∗X(logD)×X T ∗

X(logD)

ϕπ=π∗p233

p2 //

p

T ∗X(logD)

π

π∗T∗

X(logD)

OO

T ∗X(logD) π

// X

.

♣r ♦♥strt♦♥ θ ∈ H0(T ∗X(logD),Ω1

T ∗X(logD)(log(π

∗(D))) ♥ ♣♦s ω = dθ

Pr ♦♥strt♦♥ ω st ♥ ♦r♠ ♦s②♠♣tq

♦r♦r ❬t♦ ❪ ♦t D ♥ sr à r♦s♠♥ts ♥♦r♠① X.

♦♣ (T ∗X(logD), π∗(D)) st ♥ rété ♦s②♠♣tq

Pr ♦t (U0, x0) ♥ rt ♦ ♦rt♠q X ♥ x0. ①st ♥ t

♥ s ♠ ♦♥t♦♥s ξi; i = 1, ..., n ♦♦♠♦r♣s é♥s sr π∗(U0) ts

q

θ|π∗(U0) =

p∑

i=1

ξidxixi

+

n∑

i=p+1

ξidxi

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

①♠♣ ❬♦t♦ ❪ X st ♥ sr ♦♠♣① ♠♥ ♥

sr rét D t s [D] st ss ♥t♥♦♥q K∗X , ♦rs ♦♣ (X,D)

st ♥ rété ♦s②♠♣tq t ♣s ♦r♠ ♦s②♠♣tq ss♦é st

ω ∈ K([D]) ⋍ OX .

é♥t♦♥ ♥ ♣♣ ♦r♠ ♦♠ ♦rt♠q ♦♥ ♥ sr rét

t r D ♥ rété ♦♠♣① X ♠♥s♦♥ n t♦t st♦♥ s♥s ③ér♦s

ΩnX(logD)

♣rès é♦rè♠ ♥s ❬t♦ ❪ s ♦r♠s ♦♠s ♦rt♠qs s♦♥t

♦r♠

µ =1

hdz1 ∧ dz2 ∧ ... ∧ dzn

Pr♦♣♦st♦♥ ♦t D = h = 0 ♥ sr r ♥ rété ♦♠♣①

X ♠♥s♦♥ α ♥ ♦r♠ ♦♦♠♦r♣ r♠é X t µ ♥ ♦r♠ ♦♠

♦rt♠q X. ♦rs t♦t tr π X t q

iπµ = α

st P♦ss♦♥ ♦rt♠q ♦♥ D.

Pr ♦t a ♥ ♦♥t♦♥ ♦♦♠♦r♣ ♥ s♥♥♥t ♣s sr X. ♥ ♣♦s

µ =a

hdx ∧ dy ∧ dz, α = αxdx+ αydy + αzdz

Pr ♥ rt ♦♥ ♦t♥t

π =h

a(αz∂x ∧ ∂y + αy∂z ∧ ∂x + αx∂y ∧ ∂z)

P♦r ♠♦♥trr q π st P♦ss♦♥ st érr ♥tté ♦ q ♥s

s s rés♠ à

π12(∂yπ23 − ∂xπ31) + π13(∂zπ23 − ∂yπ12) + π23(∂zπ31 − ∂xπ12) = 0

♦ù π12 = ha−1αz, π13 = −ha−1αy t π23 = ha−1αx.

r

π12(∂yπ23 − ∂xπ31) = h2a−1αz(αx∂ya−1 − αy∂xa

−1) + ha−2αz(αx∂yh − αy∂xh) +

a−2h2αz(∂yαx − ∂xαy)

π13(∂z(π31)−∂x(π12)) = h2a−1αy(αz∂xa−1−αx∂za−1)+ha−2αy(αz∂xh−αx∂zh)+

h2a−2αy(∂xαz − ∂zαx)

π23(∂z(π31)−∂y(π12)) = h2a−1αx(αy∂za−1−αz∂ya−1)+ha−2αx(αy∂zh−αz∂yh)+

a−2h2αx(∂zαy − ∂yαz)

t dα = 0 s t s♠♥t s

∂yαx − ∂xαy = ∂xαz − ∂zαx = ∂zαy − ∂yαz = 0

s♣ s SU(2) ♠♦♥♦♣ôs ♠♥étqs r

q ♠♦♥tr q π st ♥ t♥sr P♦ss♦♥ Pr rs ♣♦r t♦t st♦♥ f

OX ♦♥

f,− = a−1h[(∂xfαz − ∂zfαx)∂y + (∂zfαy − ∂yfαz)∂x + (∂yfαx − ∂xfαy)∂z]

st ♥ ért♦♥ ♦rt♠q ♦♥ hOX .

s♣ s SU(2) ♠♦♥♦♣ôs ♠♥étqs r

♣rès ♦♥s♦♥ ♥s ❬♦♥s♦♥ ❪ s♣ ♠♦ ér M2 s

♠♦♥♦♣ôs ♠♥étqs r st ♥ t♦♥ rété ♦♠♣① R2

s ♦♥t♦♥s rt♦♥♥s w(z) =f(z)

g(z) ré ts q w(∞) = 0.

♥ ét♥t ②♥♠q s ♠♦♥♦♣ôs ♦♥ ♣r♦ ♥s

❬t② t♥ ❪ q s β1 t β2 s♦♥t s r♥s g, ♦rs

ω =1

f(β1)f(β2)(f(β2)df(β1) ∧ dβ1 + f(β1)df(β2) ∧ dβ2)

st ♥ strtr s②♠♣tq sr M2.

♦s ♦♥s ♠♦♥trr q ω st ♥ ♦r♠ ♦s②♠♣tq ♦♥ D =

R(f, g) = 0 ♦ù R(f, g) és♥ rést♥t f t g. Pr st ♥♦s ♦♥strs♦♥s

strtr P♦ss♦♥ ♥t ♣r ω t ♥♦s ♠♦♥tr♦♥s q st ♦rt♠q

♦♥ D.

♣rès ❬♦♥s♦♥ ❪ s éé♠♥ts R2 s♦♥t s♦s ♦r♠

f(z)

g(z)=

a0 + a1z

b0 + b1z + z2

r w(∞) = 0. ♥s s

w(z) =f(z)

g(z)=

a0 + a1z

b0 + b1z + z2

t ♦♥

g= b21 − 4b0, β1 = −1

2

(b1 +

√g

), β2 =

1

2

(−b1 +

√g

).

♥ ♥ ét q

dβ1 =1√g

(β1db1 + db0) , dβ2 = − 1√g

(β2db1 + db0)

df(β1) ∧ dβ1 =1√g

(β1da0 ∧ db1 + da0 ∧ db0 + β21da1 ∧ db1 + β1da1 ∧ db0

)

df(β2) ∧ dβ2 = − 1√g

(β2da0 ∧ db1 + da0 ∧ db0 + β22da1 ∧ db1 + β2da1 ∧ db0

).

♣tr r s strtrs P♦ss♦♥ ♦rt♠qs

♥ ♣♦s

ω =df(β1)

f(β1)∧ dβ1 +

df(β2)

f(β2)∧ dβ2.

Psq f t g ♥♦♥t ♣s r♥s ♦♠♠♥s ω st ♥ é♥ sr C4. ♥s

s ①♣rss♦♥ ♥t

√gf(β1)f(β2)ω

= (β1f(β2)− β2f(β1)) da0 ∧ db1 + (f(β2)− f(β1)) da0 ∧ dbo+

(β21f(β2)− β22f(β1)

)da1 ∧ db1 + (β1f(β2)− β2f(β1)) da1 ∧ db0.

r

f(β1)f(β2) = (a0 + a1β1)(a0 + a1β2) = a20 + a1a0(β1 + β2) + a21β1β2= a20 − a1a0b1 + a21b0.

β1f(β2)− β2f(β1) = −a0√g

β21f(β2)− β22f(β1) = (a0b1 − a1b0)√g

f(β2)− f(β1) = a1√g.

Pr rs ∣∣∣∣∣∣

a0 a1 0

0 a0 a1b0 b1 1

∣∣∣∣∣∣= a20 − a1a0b1 + a21b0.

s♥st q R(f, g) = f(β1)f(β2). ♥s st ♥♦s ♥♦tr♦♥s R.

♥ sstt♥t s ①♣rss♦♥s ♥s ♦♥ ♦t♥t

ω =1

R(−a0da0 ∧ db1 + a1da0 ∧ db0 + (a0b1 − a1b0)da1 ∧ db1 − a0da1 ∧ db0) .

♥ ♥ ét q

R2ω ∧ ω= (−a0da0 ∧ db1 + a1da0 ∧ db0 + (a0b1 − a1b0)da1 ∧ db1 − a0da1 ∧ db0)∧ (−a0da0 ∧ db1 + a1da0 ∧ db0 + (a0b1 − a1b0)da1 ∧ db1 − a0da1 ∧ db0)= 2

(a20da0 ∧ db1 ∧ da1 ∧ db0 + a1(a0b1 − a1b0)da0 ∧ db0 ∧ da1 ∧ db1

)

= 2(a20 − a1(a0b1 − a1b0)

)da0 ∧ da1 ∧ db0 ∧ db1

= 2Rda0 ∧ da1 ∧ db0 ∧ db1.

♦ù

ω ∧ ω =2

Rda0 ∧ da1 ∧ db0 ∧ db1 6= 0.

♥ ♦♥t q ω st ♥ ♦r♠ ♦s②♠♣tq ♦♥ D := R = 0. r♦t P♦ss♦♥ ss♦é à ω st

u, vω = f(β1)

(∂u

∂β1

∂v

∂f(β1)− ∂u

∂f(β1)

∂v

∂β1

)+f(β2)

(∂u

∂β2

∂v

∂f(β2)− ∂u

∂f(β2)

∂v

∂β2

).

s♣ s SU(2) ♠♦♥♦♣ôs ♠♥étqs r

s♥st q ♠♣ ♠t♦♥♥ ss♦é à u st

Xu = f(β1)

(∂u

∂β1

∂

∂f(β1)− ∂u

∂f(β1)

∂

∂β1

)+ f(β2)

(∂u

∂β2

∂

∂f(β2)− ∂u

∂f(β2)

∂

∂β2

).

♥ ♣♣q♥t ♠♣ sr R = f(β1)f(β2), ♦♥ ♦t♥t

Xu(R) = R

(∂u

∂β1+

∂u

∂β2

).

♦♠♠ Xu(R) st ♥ éé♠♥t é ♥♥ré ♣r R, ♦♥ ♦♥t q −,−ωst ♥ strtr P♦ss♦♥ ♦rt♠q ♦♥ D.

tr sr D = R(f, g) = 0.♣rès q ♣réè srD ♣♦r éqt♦♥ x2−xyt+y2z. ♥ r♠rq

q

x2 − xyt+ y2z = (x− yt

2)2 + y2(z − t2

4)

= X2 + Y 2Z

♦ù X = x− yt

2, Y = y t Z = z− t2

4. Pr rs X

∂h

∂X+Y

∂h

∂Y= 2h. ♥ ♥ ét

s②stè♠ ♠♥♠ é♥értrs s♥t Der(logD).

δ1 = X∂

∂X+ Y

∂

∂Y

δ2 = Y∂

∂Y+ 2Z

∂

∂Z

δ3 = Y 2 ∂

∂X+ 2X

∂

∂Z

δ4 = Y Z∂

∂X−X

∂

∂Y

Psq Der(logD) st ♥ s♦s ♠♦ Der q st r♥ ♥ ♣t êtr

r r r♥ ♥s♠ ♠♥♠ ss é♥értrs st s♣érr à

♠rq té♦r strtr P♦ss♦♥ ♦rt♠q ♦♥strt t♦t

♦♥ ♣tr été t ♣♦r s srs rs ①♠♣ s♣

s ♠♦♥♦♣ôs r ♥♦s ♠♦♥tr q ♣t ss s é♥r ♣♦r rt♥s

srs ♥♦♥ rs

♣tr

♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♦♠♠r ♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥ ♦

rt♠q

èrs ♥rt ♦rt♠qs

trtr èr ♥rt sr ΩA(log I) ♥t ♣r♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I.

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

qs strtrs èr ss♦és ① strtrs

P♦ss♦♥ ♦rt♠qs

trtrs èr ♥rt sr ΩX(logD)

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠qs s str

trs ♦s②♠♣tqs

♦♦♠♦♦ P♦ss♦♥ t P♦ss♦♥ ♦

rt♠q strtr P♦ss♦♥ x, y = 0, x, z =

0, y, z = xyz sr A = C[x, y, z]

♥tr♦t♦♥

♦s ♠♦♥trr♦♥s ♥s tt ♣rt q t♦t strtr P♦ss♦♥ ♦rt

♠q ♣r♥♣ ♥t sr ♠♦ s ér♥ts ♦r♠s ♦rt♠qs

♥ strtr èr ♥rt tt strtr é♦ ♥ r♣rés♥

tt♦♥ ♠♦ s ér♥ts ♦r♠s ♦rt♠qs ♣r s ért♦♥s

♦rt♠qs tt r♣rés♥tt♦♥ ♥t ♦♠♣① P♦ss♦♥ ♦rt♠q

♦s ♦♥s qqs r♦♣s ♦♦♠♦♦ ♦♠♣① ♦s ♠♦♥tr♦♥s q

s r♦♣s ♦♦♠♦♦ P♦ss♦♥ t P♦ss♦♥ ♦rt♠qs s strtrs

P♦ss♦♥ ♦s②♠♣tq s♦♥t s♦♠♦r♣s

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦♥strt♦♥ érq ♦♦♠♦♦ P♦s

s♦♥ ♦rt♠q

♥s tt ♣rt A és♥r ♥ èr sr ♥ ♥♥ ♦♠♠tt ♥tr

R rtérstq ③ér♦ t I ♥ é ♣r♦♣r A.

èrs ♥rt ♦rt♠qs

♦s ♣♣r♦♥s ♥♥ t♦t ♥♥ éq♣é ♥ r♦t

♦t L ♥ ♥♥ q st ♥ ♣s ♥ A♠♦ ♥ é♥t♦♥ s♥t

é♥t♦♥ ❬♥rt ❪ ♥ ♣♣ strtr èr ♥rt

♥s L t♦t ♦♠♦♠♦r♣s♠ A♠♦s t èrs ρ : L → DerAs♦♠s à ♦♥t♦♥ ♦♠♣tté s♥t

[α, aµ] = ρ(α)(a)µ+ a[α, µ]

♥s st ♥♦s ♣♣r♦♥s èr ♥rt t♦t tr♣t (L, [−,−], ρ)

♦r♠é ♥ ♥♥ (L, [−,−]) q ♥ ♣s st ♥ A♠♦ t ♥ strtr

èr ♥rt ρ sr L.

Pr s♦ rté t♦t èr ♥rt (L, [−,−], ρ) sr r♣rés♥té ♣r

L. P♦r t♦s µ ∈ L t a ∈ A, ρ(µ)(a) sr ♥♦té s♠♣♠♥t µ(a).

♦♥t P,Q ① A♠♦s ①st ① ç♦♥s ♠ttr ♥ strtr

A♠♦ sr r♦♣ t HomR(A,B) à s♦r

r : A×HomR(P,Q) → HomR(P,Q), ra()(p) := r(a,)(p) := (a+)(p) = (ap)

t

l : A×HomR(P,Q) → HomR(P,Q), la()(p) := (a,)(p) := (a)(p) = a(p)

P♦r t♦t a ∈ A t ∈ HomR(P,Q), ♦♥ ♣♦s

δa := ra()− la().

Pr ♦♥strt♦♥ δa st ♥ ♥♦♠♦r♣s♠ R♥ér HomR(P,Q). ♦♥ ♣♦r

t♦t a, b ∈ A ♦♠♣♦sé δa δb st ♥ é♥

st rss ♥s ❬rss ❪ t ❱♥♦r♦ ♥s

❬❱♥♦r♦ ❪ ♦♥ ♦♣t é♥t♦♥ s♥t

é♥t♦♥ : P → Q st ♣♣é ♦♣értr ér♥t sr A ♦rr ♥érr

♦ é à s s st t t s ♣♦r t♦s a0, ..., as ∈ A ♦♥

δa0 δa1 ... δas() = 0

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♥ r♠rq q ♥s♠ s ♦♣értrs ér♥ts sr A ♦rr ≦ s ♦r♠ ♥

r♦♣ t s t♦♥s é♥s ♣r s rt♦♥s t ♦♥ ♥ t

① ♠♦s sr A à s♦r

s(P,Q) ♣♦r t♦♥ r t +s (P,Q) ♣♦r l. ♥ ♥♦t (+)s (P,Q) ♠♦

♦t♥ ♥ ♦♥♥t s ① t♦♥s

Pr s♦ s♠♣té +1 (P ) és♥r +1 (P, P ) ♣♦r t♦t A♠♦ P.

♥ s strt♦♥ ♣♦r t♦t ∈ s(P,Q), ♦♥

• P♦r s = 0 :

0 = δa()(p) = (ap)− a(p)

♣♦r t♦s a ∈ A t p ∈ P. ♦♥ s ♦♣értrs ér♥ts ♦rr ③ér♦ s♦♥t

①t♠♥t s ♣♣t♦♥s ♥érs P rs Q.

• P♦r s = 1 :

0 = (δab())(p) = δa((bp)− b(p)) = (abp)− b(ap)−a(bp)+ab(p).

tr♠♥t t s éé♠♥ts 1(P,Q) ér♥t rt♦♥

(abp)− b(ap)− a(bp) + ab(p) = 0.

♥s s ♦♣értrs ér♥ts ♦rr ≦ 1 A rs Q s♦♥t rtérsés

♣r rt♦♥

(ab)− b(a)− a(b) + ab(1) = 0

♣♦r t♦t a, b ∈ A. st ♣r♥r p = 1 ♥s ♣♣♦♥s ss

q♥ ért♦♥ A à rs ♥s Q st ♥ éé♠♥t HomR(A, Q)

ér♥t

(ab) = a(b) + b(a)

♣♦r t♦t a, b ∈ A. ♥ ♥♦t Der(A, Q) ♥s♠ s ts ért♦♥s P♦r

t♦t ∈ Der(A, Q), ♦♥

(δab())(1) = (ab)−b(a)−a(b)+ab(1) = (ab)−b(a)−a(b) = 0

♣♦r t♦t a, b ∈ A. ♦♥ Der(A, Q) st ♥ s♦s ♠♦ strt 1(A, Q).

été ♥trs s ① ♠♦s ②♥t ♦rsq (1) = 0 ♣♦r t♦t

∈ 1(A, Q).

♦t P ♥ Aèr t L ♥ P ♠♦ éq♣é ♥ strtr é♥

♣r ♥ r♦t [−,−].

é♥t♦♥ ♥ ♣♣ strtr P èr ♥rt sr L t♦t ♦♠♦

♠♦r♣s♠ A♠♦s ρ : L → 1(P, P ) stss♥t ♣r♦♣rété ♦♠♣t

té s♥t

[α, pµ] = ρ(α)(p)µ+ p[α, µ]

♣♦r t♦t α, µ ∈ L t p ∈ P.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦♠♠ ♥s s s èrs ♥rt ♥ P èr ♥rt

st ♥ qr♣t (L, [−,−], ρ, P ) ♦ù ρ st ♥ strtr P èr

♥rt sr L. ♦rsq♥ ♦♥s♦♥ ♥st ♣♦ss t♦t P èr

♥rt (L, [−,−], ρ) sr ♥♦té s♠♣♠♥t L.

♥ ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦t èr ♥rt sr A st ♥ Aèr

♥rt

Pr é♦ t q DerA st ♥ s♦s ♠♦ 1(A,A).

♦s ♥ és♦♥s q s èrs ♥rt s♦♥t ♥ s ♣rtr P

èrs ♥rt Pr rs t♦t strtr P èr ♥rt

sr L ♥t s r♣rés♥tt♦♥ ♣r s ♦♣értrs ér♥ts ♦rr sr A. ♥

♣t ♦♥ é♥r ♥ ♦♦♠♦♦ ss♦é à tt r♥èr

♦t L ♥ P èr ♥rt t q ♥ ♥tr ♥tr

é♥t♦♥ ♥ ♣♣ P ♦♥ ♠♥s♦♥ q ♦ q−P ♦♥ ss♦é à

ρ t♦t ♣♣t♦♥ q♥ér tr♥é L rs P.

♥ ♥♦tr Ltq(L,P ) s♣ s q − P ♦♥s Pr é♥t♦♥ ♦♥

Lt0(L,P ) = P.

♥ é♥t ♥ ♣♣t♦♥ ♥ér dρ : Ltq(L,P ) → Ltq+1(L,P ) ♣r ♦r♠

(dρf)(x1, ..., xq+1)

=q+1∑i=1

(−1)i+1ρ(xi)f(x1, ..., xi, ..., xq+1)

+q+1∑i<j

(−1)i+j+1f([xi, xj ], x1, ..., xi, ..., , xj , ..., xq+1)

Pr♦♣♦st♦♥ ♣♣t♦♥ dρ ér

dρ dρ = 0

Pr ♦s ♣r♦♣♦s♦♥s ♥ é ♣r ♦s réér♦♥s tr à ♥♥①

♣♦r ♥ ♣r ♦♠♣èt t été

• P♦r q = 1, ♦♥ ♣♦s = dρf ♣♦r t♦t f ∈ P. ♦rs ♣♦r t♦t x ∈ L ♦♥

(x) = ρ(x)f.

Pr rs

(dρg)(x1, x2) = ρ(x1)g(x2)− ρ(x2)g(x1)− g([x1, x2]).

♥ r♠♣ç♥t g ♣r , ♦♥ ♦t♥t

dρ dρ(f)(x1, x2) = ρ(x1)(x2)− ρ(x2)(x1)−([x1, x2])

= ρ(x1)ρ(x2)f − ρ(x2)ρ(x1)f − ρ([x1, x2])f

= ([ρ(x1), ρ(x2)]− ρ([x1, x2])) f

= 0

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

• P♦r q = 3, ♦♥ ♣♦s

g(x1, x2) = (dρf)(x1, x2) = ρ(x1)f(x2)− ρ(x2)f(x1)− f([x1, x2]).

P♦r t♦t f ∈ Lt1(L,P ).Pr rs ♣♦r t♦s x1, x2, x3 ∈ L, ♦♥

(dρg)(x1, x2, x3) = ρ(x1)g(x2, x3)− ρ(x2)g(x1, x3) + ρ(x3)g(x1, x2)

−g([x1, x2], x3) + g([x1, x3], x2)− g([x2, x3], x1).

♥ r♠♣ç♥t g ♣r s♦♥ ①♣rss♦♥ ♦♥ ♦t♥t

(dρg)(x1, x2, x3)

= ρ(x1) (ρ(x2)f(x3)− ρ(x3)f(x2)− f([x2, x3]))

−ρ(x2) (ρ(x1)f(x3)− ρ(x3)f(x1)− f([x1, x3]))

+ρ(x3) (ρ(x1)f(x2)− ρ(x2)f(x1)− f([x1, x2]))

−ρ([x1, x2])f(x3) + ρ(x3)f([x1, x2]) + f([[x1, x2], x3])

+ρ([x1, x3])f(x2)− ρ(x2)f([x1, x3])− f([[x1, x3], x2])

−ρ([x2, x3])f(x1) + ρ(x1)f([x2, x3]) + f([[x2;x3], x1])

♥ t♦rs♥t s trs f(x3), f(x2) t f(x1) rs♣t♠♥t ♦♥ ♦t♥t

(dρg)(x1, x2, x3)

= (ρ(x1)ρ(x2)− ρ(x2)ρ(x1)− ρ([x1, x2])) f(x3)

+ (−ρ(x1)ρ(x3) + ρ(x3)ρ(x1) + ρ([x1, x3])) f(x2)

+ (ρ(x2)ρ(x3)− ρ(x3)ρ(x2)− ρ([x2, x3])) f(x1)

+f ([[x1, x2], x3]− [[x1, x3], x2] + [[x2;x3], x1])

+ρ(x1)(f([x2, x3])− f([x2, x3])) + ρ(x2)(f([x1, x3])

−f([x1, x3])) + ρ(x3)(f([x1, x2])− f([x1, x2]))

été ré é♦ ♥tté ♦ r♦t [−,−] t t

q ρ ♦♠♠t s r♦ts

é♥t♦♥ ♦♦♠♦♦ ♦♠♣①

... // Lt∗+1(L,P ) // Lt∗(L,P ) // ...

st ♣♣é ♦♦♠♦♦ ♥rt L à rs ♥s P.

♥ s sss s ♣s ♠♣♦rt♥t èrs ♥rt (L, ρ) st

♣♦r q ρ : L → DerA st ♥ ♠♦♥♦♠♦r♣s♠ èrs ♦♥

r♠rqr q♥ é♥ér ♥ s♦s ♥s♠ L DerA ♠♥ ♥s♦♥ st ♥

èr ♥rt s t s♠♥t s st ♥ s♦s ♠♦ DerA.

♥s DerA(log I) t DerA s♦♥t s èrs ♥rt ♦♦♠♦♦

♥rt DerA(log I) rs♣ DerA st ♦♦♠♦♦ ♠

♦rt♠q A.

♥s st I és♥ ♥ é A ♥♥ré ♣r S = u1, ..., up ⊂ A.♦t (L, ρ) ♥ èr ♥rt sr A. ♥ ♠♠ s♥t

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♠♠ ρ(L)∩DerA(log I) st ♥ s♦s èr ♥♦♥ tr DerA.

Pr ρ ét♥t ♥ ♦♠♦♠♦r♣s♠ èrs ρ(L)∩DerA(log I) st r♠é

♣♦r r♦t DerA. Pr rs ♣♦r t♦t l ∈ L, u ∈ S uρ(l) = ρ(ul) ∈ρ(L) ∩DerA(log I).

é♥t♦♥ ❯♥ èr ♥rt ♦rt♠q ♦♥ I st ♥ tr♣t

(L, [−,−], ρ, I) ♦r♠é ♥ A♠♦ L éq♣é ♥ r♦t [−,−] t ♥

♦♠♦♠♦r♣s♠ èr ρ : L→ DerA(log I) stss♥t

♦t (L, [−,−], ρ, I) ♥ èr ♥rt ♦rt♠q ♦♥ I P♦rt♦t x, y, z ∈ L, a ∈ A, ♦♥

(ρ[x, y]− [ρ(x), ρ(y)])(a).z

= ρ[x, y](a).z − [ρ(x), ρ(y)](a).z

= ρ[x, y](a).z − ρ(x)[ρ(y)(a)].z + ρ(y)[ρ(x)(a)].z

= [[x, y], az]− a[[x, y], z]− [x, ρ(y)(a).z] + ρ(y)(a)[x, z] + [y, ρ(x)(a)z]− ρ(x)(a)[y, z]

= [[x, y], az]− a[[x, y], z]− [x, [y, az]] + [x, a[y, z]]+

+[y, a[x, z]]− a[y, [x, z]] + [y, [x, az]]− [y, a[x, z]]− [x, a[y, z]] + a[x, [y, z]]

= − ([az, [x, y]] + [x, [y, az]] + [y, [x, az]])− a ([[x, y], z] + [[y, z], x] + [[z, x], y])

= 0.

♥s (ρ[x, y] − [ρ(x), ρ(y)])(a) = 0 ♣♦r t♦t a ∈ A s A st s♥s t♦rs♦♥ ♦♥

ρ[x, y] = [ρ(x), ρ(y)]. st ♣r ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦t L ♥ A♠♦ s♥s t♦rs♦♥ Ann(L) = 0

❯♥ ♦♠♦♠♦r♣s♠ ρ : L → DerA(log I) A♠♦s st ♥ strtr èr

♥rt ♦rt♠q s t s♠♥t s stst

♦t P ♥ A♠♦ ♥ ♣♦s DerA(log I, P ) = δ ∈ DerA(A, P )t q δ(u) ∈uP ; ♣♦r t♦tu ∈ S. ♥ é♥t♦♥ s♥t

é♥t♦♥ DerA(log I, P ) st ♣♣é ♠♦ s ért♦♥s A ♦rt

♠qs ♣r♥♣s ♦♥ I à rs ♥s P.

s♥st q DerA(log I) = DerA(log I,A).

♦t ∈ +1 (P ); ♣♦r t♦t a, b ∈ A, p ∈ P, ♦♥

(r(a+ b)− l(a+ b))p = (ap)− a(p) +(bp)− b(p)

s♥ st q ♥t ♥ ♠♦r♣s♠ r♦♣s σ : A → HomR(P,Q) a 7→δa = r(a)− l(a).

Pr♦♣♦st♦♥ P♦r t♦t ∈ +1 (P ) t t♦t A♠♦ Q,

σ ∈ Der(A,HomR(P,Q)).

Pr Psq ∈ +1 (P ), ♣♦r t♦s a, b ∈ A, p ∈ P ♦♥

(abp) = b(ap) + a(bp)− ab(p).

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♦♥

(abp)− ab(p) = (bσ(a) + aσ(b))(p).

stàr

σ(ab) = aσ(b) + bσ(a)

♦ù réstt

♥s♦♥ s ért♦♥s ♦rt♠qs ♦♥ I ♥s DerA ♣r♠t ♥

sr s ♦♣értrs ér♥ts ts q σ ∈ DerA(log I).P♦s♦♥s

+1 (log I) = ∈ +1 (P )|σ ∈ DerA(log I).

+1 (log I) st ♥♦♥ tr r ♣♦r t♦t ∈ +1 (P ), u ∈ S, ♦♥ u ∈+1 (log I). Pr rs +1 (log I) ♣♦ssè ♣r♦♣rété s♥t

Pr♦♣♦st♦♥ +1 (log I) st ♥ èr ♥rt ♦rt♠q

♦♥ I.

Pr ♣rès q ♣réè ①st ♥ ♣♣t♦♥

σ : +1 (log I) → DerA(log I) 7→ σ

P♦r t♦t f ∈ A, s ∈ P ♦♥ σf = fσ t

σ[ϕ1,ϕ2](f)s = [ϕ1, ϕ2](fs)− f [ϕ1, ϕ2](s)

= ϕ1ϕ2(fs)− ϕ2ϕ1(fs)− fϕ1ϕ2(s) + ϕ2ϕ2(s)

= ϕ1(σϕ2(f)s+ fϕ2(s))− ϕ2(σϕ1(f)s+ fϕ1(s))− f [ϕ1, ϕ2]s

= ϕ1(σϕ2(f)s) + ϕ1(fϕ2(s))− ϕ2(σϕ1(f)s)− ϕ2(fϕ1(s))− f [ϕ1, ϕ2]s

= σϕ1(σϕ2(f))s+ σϕ2(f)ϕ1(s) + σϕ1(f)ϕ2(s) + fϕ1(ϕ2(s))−σϕ2(σϕ1(f))s− σϕ1(f)ϕ2(s)− σϕ2(f)ϕ1(s)− fϕ2(ϕ1(s))− f [ϕ1, ϕ2]s

= [σϕ1 , σϕ2 ](f)s

tr ♣rt ϕ1, ϕ2 ∈ +1 (log I), f ∈ A t s ∈ P ♥♦s ♦♥s

[ϕ1, fϕ2] = ϕ1(fϕ2(s))− (fϕ2)(ϕ1(s))

= fϕ1(ϕ2(s)) + σϕ1(f)(ϕ2(s))− fϕ2(ϕ1(s))

= σϕ1(f)(ϕ2(s)) + f [ϕ1, ϕ2]

♥ ♣♦s

+1 (log I, P ) := ∈ +1 (A, P );σ ∈ DerA(log I, P ). ♦rs +1 (log I, P )st rtérsé ♣r

é♦rè♠ P♦r t♦t ∈ HomR(A, P ) s ♣r♦♣rétés s♥ts s♦♥t éq

♥ts

∈ +1 (log I, P )

σ ∈ Der(log I,HomR(A, P ))

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

Pr ♦♥t a, b ∈ A t ∈ HomR(A, P ).

σ(u) ∈ uHomR(A, P )t

σ(ab) = aσ(b) + bσ(a)

♦rs ♣♦r t♦t p ∈ P,

(abp) = a(bp) + b(ap)− ab(p)

Pr rs

δa,b(p) = (abp)− a(bp)− b(ap) + ab(p)

q ♠♣q ♣rès éqt♦♥ q δa,b = 0. ♦♥ st ♥ ♦♣értr

ér♥t ♦rr ≤ 1.

ré♣r♦q é♦ é♥t♦♥ +1 (log I, P ).

♥ ♥♦t

DI(B ⊂ P ) := ∈ DerA(log I, P )|(A) ⊂ BPr é♥t♦♥ DI(B ⊂ P ) = DerA(log I, B) s B st ♥ s♦s ♠♦ P.

♥ é♥t ♣r ♥t♦♥ ♥ st DIi (P ) ⊂ +

1 (log I, P ), i ≧ 0 A♠♦s

♥ ♣♦s♥t DI0 (P ) = P,DI

1 (P ) = DerA(log I, P ) t DIi+1(P ) = DI(DI

i (P ) ⊂((+

1 )i(log I, P ))) ♦ù (+

1 )i(P ) = +

1 (...(+1 (log I, P )...).

é♥t♦♥ s éé♠♥ts DIi (P ) s♦♥t ♣♣és ♣♦②ért♦♥s A ♥s P

♦rt♠qs ♦♥ I. ♣r♦♣♦st♦♥ s♥t ♦♥♥ ♥ sr♣t♦♥ été DI

i (P )

Pr♦♣♦st♦♥ P♦r t♦t i ≧ 1 ♥ éé♠♥t ∈ HomK(A, DIi−1) ♣♣rt♥t à

DIi s t s♠♥t s ♣♦r t♦t a, b ∈ A, stst s ♣r♦♣rétés s♥ts

(ab) = a(b) + b(a)

(a, b) +(b, a) = 0

Pr st ♥tq à ♣r♦♣♦sé ♥s ❬rss ❪

rss♦rt ♠♠ q DIi (A) t Hom(

i∧Ω(log I,A)) s♦♥t s♦♠♦r♣s

trtr èr ♥rt sr ΩA(log I) ♥t ♣r♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I.

♥s tt ♣rt ♥♦s ♦♣t♦♥s s ♥♦tt♦♥s st♦♥ ♥ s♣♣♦sr

♥ ♦tr q A st éq♣é ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣

♦♥ ♥ é I ♥♥ré ♣r S t ♥♦s és♥r♦♥s ♦r♠ ss♦é ♣r ω.

ér♥t d sr s♠♣♠♥t ♥♦té d t ♦♠♣① ss♦é

0d // A d // 1∧

ΩA(log I)d // ... d // i∧

ΩA(log I)d // i∧

ΩA(log I)d // ...

sr ♣♣é ♦♠♣① ♠ ♦rt♠q A.

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

trtr ♥t sr ΩA(log I)

♣r♦♣♦st♦♥ s♥t ♦♠♣èt s ♣r♦♣rétés éré s ér

t♦♥s ♦rt♠qs

Pr♦♣♦st♦♥ ♦t δ ∈ DerR(log I). P♦r t♦tx

u∈ S−1ΩR(A), ♦♥

Lδ(x

u) =

1

uLδ(x)−

δ(u)

u

x

u

Pr ♥ t ♣♦r t♦t x ∈ ΩA, u ∈ I∗,Lδ(x) = Lδ(ux

u) = uLδ(

x

u) + δ(u)

x

u.

♦♥ Lδ(x

u) =

1

uLδ(x)−

δ(u)

u

x

u.

râ à tt ♣r♦♣♦st♦♥ ♥♦s ♦♥s s ♣r♦♣rétés s♥ts

♦r♦r P♦r t♦t u ∈ S t δ ∈ DerK(log I) ♦♥

Lδ(du

u) = d(

δ(u)

u)

Pr ♦tdu

u∈ ΩA(log I). ♦s és♦♥s Pr♦♣♦st♦♥ q

Lδ(d(u)

u) =

1

uLδ(d(u))−

δ(u)

u

d(u)

u

=1

ud(δ(u))− δ(u)

u

d(u)

u

Psq δ ∈ DerA(log I), ①st ♦♥ c ∈ A∗ t q δ(u) =

uc. s♥st qdδ(u)

u= d(c) +

d(u)

u= d(

δ(u)

u) +

δ(u)

u

d(u)

ut ♦♥

Lδ(d(u)

u) = d(

δ(u)

u) +

δ(u)

u

d(u)

u− δ(u)

u

d(u)

u= d(

δ(u)

u)

♦r♦r P♦r t♦t strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥

I sr A ♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é H ♦♥

LH(d(u)

u)

(d(v)

v) = d

(1

uvu, v

)

♣♦r t♦t u, v ∈ S.

Pr ♦♥t u, v ∈ S. ♣rès é♥t♦♥ H, ♦♥

H(d(u)

u) =

1

uH d(u) = 1

uu,− =: ϕ

♥ ♣♣q♥t ♣r♦♣♦st♦♥ ♦♥ ♦t♥t

LH(d(u)

u)

(d(v)

v

)= Lϕ

d(v)

v

= d

(ϕ(v)

v

)

= d

(1

uvu, v

)

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♣r♦♣♦st♦♥ s♥t ♦♥♥ s ①♣rss♦♥s éré s ér♥ts

♦r♠s ♦rt♠qs ♦r♠ qdu

u ♦♥ s ért♦♥s ♦rt♠qs

♣r♥♣s

Pr♦♣♦st♦♥ ♦t H ♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é à

♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I ♦r♠ ss♦é

ω. P♦r t♦t a ∈ A t u, v ∈ S, ♦♥

LH(a

d(u)

u)

(d(v)

v) = ad(

1

uvu, v) + 1

uvu, vd(a)

LH(a

d(u)

u)

(bd(v)

v) =

1

uu, bd(v)

v+

b

uvu, vd(a) + bad(

1

uvu, v)

LH(b

d(v)

v)

(ad(u)

u) =

b

vv, ad(u)

u+

a

uvv, ud(b) + abd(

1

uvv, u)

d

(ω(a

d(u)

u, bd(v)

v)

)= abd

(1

uvu, v

)+

b

uvu, vd(a) + a

uvu, vd(b)

Pr tt ♣r é♦ ♦r♦r ♦s r♥♦②♦♥s à ♥♥①

tt ♣r♦♣♦st♦♥ ♦♥ ét s réstts

♦r♦r ♦♥t a, b ∈ A t u, v ∈ S. ♥

−dω(aduu, bdv

v) + L

H(adu

u)

(bdv

v)− L

H(bdu

u)

(adv

v) =

=a

uu, bd(v)

v+b

va, vdu

u+ abd(

1

uvu, v

Pr ♦s és♦♥s ♣r♦♣♦st♦♥ t ♦r♦r q

−dω(aduu, bdv

v) + L

H(adu

u)

(bdv

v)− L

H(bdu

u)

(adv

v) =

= −abd[ 1uv

u, v]− b

uvu, vda− a

uvu, vdb+ a

uu, bdv

v+

b

uvu, vda+

+abd(1

uvu, v) + b

va, vdu

u+

a

uvu, vdb+ abd(

1

uvu, v).

♣rès s♠♣t♦♥ ♦♥ ♦t♥t

−dω(aduu, bdv

v) + L

H(adu

u)

(bdv

v)− L

H[bdu

u]

(adv

v)

=a

uu, bdv

v+b

va, vdu

u+ abd(

1

uvu, v

♦r♦r s♥t ♥♦s ♣r♠t rtr♦r ①♣rss♦♥ é♥ér r♦t

P♦ss♦♥ ♥t sr ΩA.

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♦r♦r ♦♥t a, b ∈ A t u, v ∈ S. ♥

−dω(adu, bdv)+LH(ad(u))(bdv)−LH(bdu)(adv) = au, bdv+ba, vdu+abd(u, v)

Pr tt été rést s ♣r♦♣rétés s♥ts

d[ω(adu, bdv)] = au, vdb+ bu, vda+ abd[u, v]. LH(adu)(bd(v)) = abd[u, v] + au, bdv + bu, vda LH(bdv)(ad(u)) = abd[v, u] + bv, adu+ av, udb

♥ ♥ ét é♠♥t q

♦r♦r ♦♥t a, b ∈ A t u, v ∈ S. ♥

−dω(aduu, bdv) + L

H(adu

u)

(bdv) − LH(bdu)(adv

v) =

a

uu, bd(v) + ba, vdu

u+

abd(1

uu, v

Pr

P♦r t♦t a, b ∈ A t u, v ∈ S ♦♥

d(ω(adu

u, bdv)) = d(

ab

uu, v)

= [1

uu, v]d(ab) + abd(

1

uu, v)

=a

uu, vdb+ b

uu, vd(a) + abd(

1

uu, v)

LH[a

du

u]

(bdv) = aLH[d(u)

u]

(bdv) + σ(H[du

u])(bdv)d(a)

= a(bLH(du

u)

(d(v)) + H(du

u)(b)dv) +

b

uu, vd(a)

= abd(1

uu, v) + a

uu, bd(v) + b

uu, vd(a)

LH[(bdv)](adu

u) = bLH(dv)(a

du

u) + σ(H(d)(v))(a

du

u)d(b)

= b(aLH(dv)(du

u) + H(dv)(a)

du

u) +

a

uv, ud(b)

= bad(1

uv, u) + bv, adu

u+a

uv, ud(b)

s♥st q

−dω(aduu, bd(v)) + L

H(adu

u)

(bdv)− LH(bdu)(adv

v)

=a

uu, bdv + ba, vd(u)

u+ abd

(1

uu, v

)

♦t S ♥ ♣rt ♠t♣t ♥ èr P♦ss♦♥ S. ♠♠ ss♦s

♠♦♥tr q ♦sé S−1A ért ♥♦♥q♠♥t ♥ strtr P♦ss♦♥ ♥t

♣r A

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♠♠ ♦t A ♥ èr P♦ss♦♥ P♦r t♦t ♣rt ♠t♣t

S ⊂ A, ♦sé S−1A ♣♦ssè ♥ strtr ♥♦♥q èr P♦ss♦♥

Pr és♥♦♥s ♣r −,− strtr P♦ss♦♥ sr A. ♦rs r♦t

a1s−11 , a2s

−12 = a1, a2(s1s2)−1 − a1, s2a2(s1s22)−1−

s1, a2a1(s21s2)−1 + a1a2s1, s2(s21s22)−1

st s♦♥ ♥q ♣r♦♦♥♠♥t sr S−1A.P♦s♦♥s

[α, β]ω = −dω(α, β) + LH(α)β − LH(β)α

♦rs [−,−]ω st R ♥ér ♥ts②♠étrq

s réstts ss♦s ①♣t♥t [−,−]ω sr s é♥értrs ΩA(log I).

♠♠ ♦t a, b ∈ A t u, v ∈ S

[adu

u, bdv

v

]

ω

=a

uu, bdv

v+b

va, vdu

u+ abd(

1

uvu, v)

[adu

u, bdv

]

ω

=a

uu, bdv + ba, vdu

u+ abd(

1

uu, v)

[adu, bdv]ω = au, bdv + ba, vdu+ abd(u, v)

[adu, b

dv

v

]

ω

= au, bduu

+b

va, vdu+ abd(

1

vu, v)

Pr s ♣r♦♣rétés s♦♥t ♥ ♦♥séq♥ rt Pr♦♣♦st♦♥ t

ss ♦r♦rs

P♦r q st ♣r♦♣rété ♦♥ ♣♦r t♦t a, b ∈ A t u, v ∈ S, ♦♥

d(ω(adu

u, bdv

v)) = abd(

1

uvu, v) + a

uvu, vdb+ b

uvu, vda

LH(a

du

u)

(bdv

v) =

a

uu, bdv

v+

b

uvu, vda+ abd(

1

uvu, v)

LH(b

dv

v)

(adu

u) =

b

vv, adu

u+

a

uvv, udb+ abd(

1

uvv, u)

t ♦♥

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

−d(ω(aduu, bdv

v)) + L

H(adu

u)

(bdv

v)− L

H(bdv

v)

(adu

u)

= −abd( 1

uvu, v)− a

uvu, vdb− b

uvu, vda+ b

vv, adu

u

+a

uu, bdv

v+

b

uvu, vda+ abd(

1

uvu, v)− b

vv, adu

u

+ − a

uvv, udb− abd(

1

uvv, u)

=b

vv, adu

u+a

uu, bdv

v+ abd(

1

uvu, v)

+ [−abd( 1

uvu, v) + abd(

1

uvu, v)] + [− a

uvu, vdb− a

uvv, udb]

+[b

uvu, vda− b

uvu, vda]

=b

vv, adu

u+a

uu, bdv

v+ abd(

1

uvu, v)

♦ù [adu

u, bdv

v

]

ω

=a

uu, bdv

v+b

va, vdu

u+ abd(

1

uvu, v)

♠♥èr ♥♦ ♦♥ ♠♦♥tr s ♣r♦♣rétés t

♥ ♣rtr ♣♦r a = b = 1 ♦♥

♦r♦r P♦r t♦t u, v ∈ S, ♦♥

[du

u,dv

v

]

ω

= d(1

uvu, v)

[du,

dv

v

]

ω

= d(1

vu, v)

[du

u, dv

]

ω

= d(1

uu, v) [du, dv] = d(u, v).

Pr♦♣♦st♦♥ P♦r t♦t u, v, w ∈ S ♦♥ [[du

u,dv

v

]

ω

,dw

w

]

ω

+

[[dv

v,dw

w

]

ω

,du

u

]

ω

+

[[dw

w,du

u

]

ω

,dv

v

]

ω

= 0

Pr ❱♦r ♥♥①

♥s ♠ê♠ ♦t ♥♦s ♦♥s

Pr♦♣♦st♦♥ P♦r t♦t u, v ∈ S t w ∈ A ♦♥

[[du

u,dv

v

]

ω

, dw

]

ω

+

[[dv

v, dw

]

ω

,du

u

]

ω

+

[[dw,

du

u

]

ω

,dv

v

]

ω

= 0

[[du

u, dv

]

ω

, dw

]

ω

+

[[dv, dw]ω ,

du

u

]

ω

+

[[dw,

du

u

]

ω

, dv

]

ω

= 0

Pr ❱♦r ♥♥① ♣♦r ♣s ét

♦s ♦♥s à ♣rés♥t ♠♦♥trr q ♣♦r t♦s ω1 = a1du1u1

+ b1dv1, ω2 = a2du2u2

+

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

b2dv2 t ω3 = a3du3u3

+ b3dv3 ♥s ΩA(log I) ♦♥

0 =

[[ω1, ω2]ω , ω3]ω + [[ω2, ω3]ω , ω1]ω + [[ω3, ω1]ω , ω2]ω

=

[[a1du1u1

, a2du2u2

]

ω

, a3du3u3

]

ω

+

[[a1du1u1

, a2du2u2

]

ω

, b3dv3

]

ω

+

[[a1du1u1

, b2dv2

]

ω

, a3du3u3

]

ω

+

[[a1du1u1

, b2dv2

]

ω

, b3dv3

]

ω

+

[[b1dv1, a2

du2u2

]

ω

, a3du3u3

]+

[[b1dv1, a2

du2u2

]

ω

, b3dv3

]

ω

+

[[b1dv1, b2dv2]ω , a3

du3u3

]+ [[b1dv1, b2dv2]ω , b3dv3]ω

+

[[a2du2u2

, a3du3u3

]

ω

, a1du1u1

]

ω

+

[[a2du2u2

, a3du3u3

]

ω

, b1dv1

]

ω

+

[[a2du2u2

, b3dv3

]

ω

, a1du1u1

]

ω

+

[[a2du2u2

, b3dv3

]

ω

, b1dv1

]

ω

+

[[b2dv2, a3

du3u3

]

ω

, a1du1u1

]+

[[b2dv2, a3

du3u3

]

ω

, b1dv1

]

ω

+

[[b2dv2, b3dv3]ω , a1

du1u1

]+ [[b2dv2, b3dv3]ω , b1dv1]ω

+

[[a3du3u3

, a1du1u1

]

ω

, a2du2u2

]

ω

+

[[a3du3u3

, a1du1u1

]

ω

, b2dv2

]

ω

+

[[a3du3u3

, b1dv1

]

ω

, a2du2u2

]

ω

+

[[a3du3u3

, b1dv1

]

ω

, b2dv2

]

ω

+

[[b3dv3, a1

du1u1

]

ω

, a2du2u2

]+

[[b3dv3, a1

du1u1

]

ω

, b2dv2

]

ω

+

[[b3dv3, b1dv1]ω , a2

du2u2

]+ [[b3dv3, b1dv1]ω , b2dv2]ω

r ♣rès ♠♠ ♦♥

[a1du1u1

, a2du2u2

]=a1u1

u1, a2du2u2

+a2u2

a1, u2du1u1

+ a1a2d

(1

u1, u2u1, u2

)

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

t ♦♥[[a1du1u1

, a2du2u2

], a3

du3u3

]

=

[a1u1

u1, a2du2u2

, a3du3u3

]+

[a2u2

a1, u2du1u1

, a3du3u3

]+

[a1a2d

(1

u1, u2u1, u2

), a3

du3u3

]

[a1u1

u1, a2du2u2

, a3du3u3

]

=a1u1u2

u1, a2u2, a3du3u3

+a3u3

a1u1

u1, a2, u3du2u2

+a1a3u1

u1, a2d(

1

u2u3u2, u3

)

[a2u2

a1, u2du1u1

, a3du3u3

]

=a2u1u2

a1, u2u1, a3du3u3

+a3u3

a2u2

a1, u2, u3du1u1

+

a2a3u2

a1, u2d(

1

u3u1u1, u3

)

[a1a2d(

1

u1, u2u1, u2), a3

du3u3

]

=a3u3

a1a2, u3d(1

u1u2u1, u2) + a1a2

1

u1u2u1, u2, a3

du3u3

+a1a2a3d

(1

u3 1

u1u2u1, u2, u3

)

Pr rs ♣r♦♣♦st♦♥ s♥t ♦♥♥ qqs ♣r♦♣rétés rtérstqs

r♦t P♦ss♦♥ ♦rt♠q ♣r♥♣

Pr♦♣♦st♦♥ ♦♥t ui ∈ S, ai ∈ A−S i = 1, 2, 3 t −,− ♥ str

tr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ I. ♥ s ♣r♦♣rétés s♥ts

P1

u3 1

u1u2u1, u2, u3+

1

u1 1

u2u3u2, u3, u1+

1

u2 1

u3u1u3, u1, u2 = 0

Pa1u1

a2u2

u2, a3, u1du3u3

=a1a2u1u2

u2, a3, u1du3u3

+

a1u1u2

a2, u1u2, a3du3u3

− a1a2u1u22

u2, a3u2, u1du3u3

Pa3u3

a1u1

u1, a2, u1du2u2

=a3a1u1u3

u1, a2, u3du2u2

+

a3u3u1

a1, u3u1, a2du2u2

− a3a1u3u21

u1, a2u1, u3du2u2

Pa3u3

a2u2

a1, u2, u3du1u1

=a3a2u3u2

a1, u2, u3du1u1

+

a3u3u2

a1, u2a2, u3du1u1

− a2a3u3u22

a1, u2u2, u3du1u1

Pa1u1

a3u3

a2, u3, u1du2u2

=a1a3u1u3

a2, u3, u1du2u2

+

a1u1u3

a2, u3a3, u1du2u2

− a1a3u1u23

a2, u3u3, u1du2u2

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

Pa2u2

a3u3

u3, a1, u2du1u1

=a2a3u2u3

u3, a1, u2du1u1

+

a2u2u3

a3, u2u3, a1du1u1

− a2a3u2u23

u3, a1u3, u2du1u1

Pa2u2

a1u1

a3, u1, u2du3u3

=a2a1u2u1

a3, u1, u2du3u3

+

a2u2u1

a3, u1a1, u2du3u3

− a2a1u2u21

a3, u1u1, u2du3u3

P a3a11

u3u1u3, u1, a2

du2u2

=a3a1u3u1

, u3, u1a2du2u2

−a3a1u1u23

u3, u1u3, a2du2u2

− a3a1u3u21

u3, u1u1, a2du2u2

P a2a31

u2u3u2, u3, a1

du1u1

=a2a3u2u3

, u2, u3, a1du1u1

−a2a3u2u23

u2, u3u3, a1du1u1

− a2a3u3u22

u2, u3u2, a1du1u1

P a1a31

u1u2u1, u2, a3

du3u3

=a1a2u1u2

u1, u2, a3du3u3

−a1a2u1u22

u1, u2u2, a3du3u3

− a1a2u2u21

u1, u2u1, a3du3u3

Pr ❱♦r ♥♥①

s ♣r♦♣rétés ♦♥ ♦t♥t [[a1du1u1

, a2du2u2

], a3

du3u3

]+

[[a2du2u2

, a3du3u3

], a1

du1u1

]+

[[a3du3u3

, a1du1u1

], a2

du2u2

]=

a1u1u2

u1, a2u2, a3du3u3

+a3a1u3u1

u1, a2, u3du2u2

+a3u3u1

u1, a2a1, u3du2u2

+

− a3a1u21u3

u1, a2u1, u3du2u2

+a1a3u1

u1, a2d(1

u2u3u2, u3) +

a2u2u1

a1, u2u1, a3du3u3

+

+a3a2u3u2

a1, u2, u3du1u1

+a3u3u2

a1, u2a2, u3du1u1

− a3a2u3u22

a1, u2u2, u3du1u1

+

a2a3u2

a1, u2d(1

u1u3u1, u3) +

a1a2u1u2

u1, u2, a3du3u3

− a1a2u1u22

u1, u2u2, a3du3u3

+

− a1a2u21u2

u1, u2u1, a3du3u3

+a3a1u3

a2, u3d(1

u1u2u1, u2) +

a3a2u3

a1, u3d(1

u1u2u1, u2)

+a1a2a3d(1

u3 1

u1u2u1, u2, u3)+

a2u2u3

u2, a3u3, a1du1u1

+a1a2u1u2

u2, a3, u1du3u3

+a1u1u2

u2, a3a2, u1du3u3

+

− a1a2u22u1

u2, a3u2, u1du3u3

+a2a1u2

u2, a3d(1

u3u1u3, u1) +

a3u3u2

a2, u3u2, a1du1u1

+

+a1a3u1u3

a2, u3, u1du2u2

+a1u1u3

a2, u3a3, u1du2u2

− a1a3u1u23

a2, u3u3, u1du2u2

+

a3a1u3

a2, u3d(1

u2u1u2, u1) +

a2a3u2u3

u2, u3, a1du1u1

− a2a3u2u23

u2, u3u3, a1du1u1

+

− a2a3u22u3

u2, u3u2, a1du1u1

+a1a2u1

a3, u1d(1

u2u3u2, u3) +

a1a3u1

a2, u1d(1

u2u3u2, u3)

+a2a3a1d(1

u1 1

u2u3u2, u3, u1)+

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

a3u3u1

u3, a1u1, a2du2u2

+a2a3u2u3

u3, a1, u2du1u1

+a2u2u3

u3, a1a3, u2du1u1

+

− a2a3u23u2

u3, a1u3, u2du1u1

+a3a2u3

u3, a1d(1

u1u2u1, u2) +

a1u1u3

a3, u1u3, a2du2u2

+

+a2a1u2u1

a3, u1, u2du3u3

+a2u2u1

a3, u1a1, u2du3u3

− a2a1u2u21

a3, u1u1, u2du3u3

+

a1a2u1

a3, u1d(1

u3u2u3, u2) +

a3a1u3u1

u3, u1, a2du2u2

− a3a1u3u21

u3, u1u1, a2du2u2

+

− a3a1u23u1

u3, u1u3, a2du2u2

+a2a3u2

a1, u2d(1

u3u1u3, u1) +

a2a1u2

a3, u2d(1

u3u1u3, u1)

+a3a1a2d(1

u2 1

u3u1u3, u1, u2)

tt r♥èr ①♣rss♦♥ ♥♦s és♦♥s q

Pr♦♣♦st♦♥ P♦r t♦s ai ∈ A t ui ∈ S i = 1, 2, 3, ♥

[[a1du1u1

, a2du2u2

], a3

du3u3

]+

[[a2du2u2

, a3du3u3

], a1

du1u1

]+

[[a3du3u3

, a1du1u1

], a2

du2u2

]= 0

Pr ❱♦r ♥♥①Pr rs ♥♦s ♦♥s s rt♦♥s s♥ts

[[a1

du1u1

, a2du2u2

]ω, b3dv3

]

ω

=a1u1u2

u1, a2u2, b3dv3 +b3a1u1

u1, a2, v3du2u2

+

b3u1

u1, a2a1, v3du2u2

− b3a1u21

u1, a2u1, v3du2u2

+a1b3u1

u1, a2d(1

u2u2, v3)+

a2u2u1

a1, u2u1, b3dv3 +b3a2u2

a1, u2, v3du1u1

+b3u2

a1, u2a2, , v3du1u1

+

−b3a2u22

a1, u2u2, v3du1u1

+a2b3u2

a1, u2d(1

u1u1, v3) +

a2a1u1u2

u1, u2, b3dv3+

− a1a2u21u2

u1, u2u1, b3dv3 −a1a2u1u22

u1, u2u2, b3dv3 + b3a1a2, v3d(1

u1u2u1, u2)

+b3a2a1, v3d(1

u1u2u1, u2) + a1a2b3d(

1

u1u2u1, u2, v3),

[[a2

du2u2

, b3dv3]ω, a1du1u1

]

ω

=a2u2

u2, b3v3, a1du1u1

+a1a2u1u2

u2, b3, u1dv3

+a1u1u2

u2, b3a2, u1dv3 −a1a2u1u22

u2, b3u2, u1dv3 +a1a2u2

u2, b3d(1

u1v3, u1)

+b3u2

a2, v3u2, a1du1u1

+a1b3u1

a2, v3, u1du2u2

+a1u1

a2, v3b3, u2du2u2

+a1b3a2, v3d(1

u1u2u2, u1) +

a2b3u2

u2, v3, a1du1u1

− a2b3u22

u2, a1u2, v3du1u1

+a1a2u1

b3, u1d(1

u2u2, v3) +

a1b3u1

a2, u1d(1

u2u2, v3)+

a2b3a1d(1

u1 1

u2u2, v3, u1)

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

t

[[b3dv3, a1

du1u1

]ω, a2du2u2

]

ω

=b3u1

v3, a1u1, a2du2u2

+a2b3u2

v3, a1, u2du1u1

+a2u2

v3, a1b3, u2du1u1

+ b3a2v3, a1d(1

u1u2u1, u2) +

a1u1

b3, u1v3, a2du2u2

+a2a1u1u2

b3, u1, u2dv3 +a2u1u2

b3, u1a1, u2dv3 −a2a1u21u2

b3, u1u1, u2dv3+a1a2u1

b3, u1d(1

u2v3, u2) +

b3a1u1

v3, u1, a2du2u2

− b3a1u21

v3, u3u1, a2du2u2

+a1a2u2

b3, u2d(1

u1v3, u1) +

a2b3u2

a1, u2d(1

u1v3, u1)+

a1b3a2d(1

u2 1

u1v3, u1, u2)

.

râ ①qs ♥♦s ♦t♥♦♥s ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦♥t ai, v3 ∈ A t ui ∈ S i = 1, 2. ♥

[[a2

du2u2

, b3dv3]ω, a1du1u1

]

ω

+

[[a1

du1u1

, a2du2u2

]ω, b3dv3

]

ω

+

[[b3dv3, a1

du1u1

]ω, a2du2u2

]

ω

= 0

Pr ❱♦r ♥♥①

♠rq ♦♥t u1, u3 ∈ S t a1, a3, b2, v2 ∈ A ♥ [[a1

du1u1

, b2dv2, ]ω, a3du3u3

]

ω

=a1u1

u1, b2v2, a3du3u3

+a3a1u3u1

u1, b2, u3dv3 +

a3u3u1

a1, u3u1, b2dv3 − a3a1u3u21

u1, b2u1, u3dv2 +a1a3u1

u1, b2d(1

u3v2, u3) +

b2u1

a1, v2u1, a3du3u3

+a3b2u3

a1, v2, u3du1u1

+a3u3

b2, u3a1, v2du1u1

+

a3b2a1, v2d(1

u1u3u1, u3) +

a1b2u1

u1, v2, a3du3u3

− a1b2u21

u1, a3u1, v2du3u3

+

a3a1u3

b2, u3d(1

u1u1, v2) +

a3b2u3

a1, u3d(1

u1u1, v2) +

a1b2a3d(1

u3 1

u1u1, v2, u3),

[[b2dv2, a3

du3u3

]ω, a1du1u1

]

ω

=b2u3

v2, a3u3, a1du1u1

+a1b2u1

v2, a3, u1du3u3

+a1u1

v2, a3b2, u1du3u3

+ b2a1v2, a3d(1

u3u1u3, u1) +

a3u3

b2, u3v2, a1du1u1

+a1a3u3u1

b2, u3, u1dv2 +a1u3u1

b2, u3a3, u1dv2 −a1a3u23u1

b2, u3u3, u1dv2+a3a1u3

b3, u3d(1

u1v2, u1) +

b2a3u3

v2, u3, a1du1u1

− b2a3u23

v2, u3u3, a1du1u1

+a3a1u1

b2, u1d(1

u3v2, u3) +

a1b2u1

a3, u1d(1

u3v2, u3)+

a3b3a1d(1

u1 1

u3v2, u3, u1)

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

t

[[a3

du3u3

, a1du1u1

]ω, b2dv2

]

ω

=a3u3u1

u3, a1u1, b2dv2 +b2a3u3

u3, a1, v2du1u1

+

b2u3

u3, a1a3, v2du1u1

− b2a3u23

u3, a1u3, v2du1u1

+a3b2u3

u3, a1d(1

u1u1, v2)+

a1u1u3

a3, u1u3, b2dv2 +b2a1u1

a3, u1, v2du3u3

+b2u1

a3, u1a1, v2du3u3

+

−b2a1u21

a3, u1u1, v2du3u3

+a1b2u1

a3, u1d(1

u3u3, v2) +

a1a3u3u1

u3, u1, b2dv2+

−a3a1u23u1

u3, u1u3, b2dv2 −a3a1u3u21

u3, u1u1, b2dv2 + b2a3a1, v2d(1

u3u1u3, u1)

+b2a1a3, v2d(1

u3u1u3, u1) + a3a1b2d(

1

u3u1u3, u1, v2).

tt r♠rq ♥♦s és♦♥s ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ P♦r t♦t u1, u3 ∈ S t a1, a3, b2, v2 ∈ A ♦♥

[[a1

du1u1

, b2dv2, ]ω, a3du3u3

]

ω

+

[[b2dv2, a3

du3u3

]ω, a1du1u1

]

ω

+

[[a3

du3u3

, a1du1u1

]ω, b2dv2

]

ω

= 0

Pr é♦ ♥ s♠♣ ♣♣t♦♥ ♥tté ♦ t ♣r♦♣rété ♥ts②♠étr −,−. ♦♥tr♦♥s ♣r ①♠♣ q

a3a1b2d(1

u3u1u3, u1, v2)

+a3b3a1d(1

u1 1

u3v2, u3, u1) + a1b2a3d(

1

u3 1

u1u1, v2, u3)

= 0

♣♣♦♥s t♦t ♦r q

1

u3 1

u1u1, v2, u3 =

1

u3u1u1, v2, u3 −

1

u3u21u1, u3u1, v2

♣s

1

u1 1

u3v2, u3, u1 =

1

u1u3v2, u3, u1 −

1

u1u23v2, u3u3, u1

t ♣s

1

u3u1u3, u1, v2 =

1

u1u3u3, u1, v2 −

1

u1u23u3, u1u3, v2+

− 1

u3u21u1, v2u3, u1.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

s♥ st ♦♥ q

a1b2a3d(1

u3 1

u1u1, v2, u3) + a3a1b2d(

1

u3u1u3, u1, v2) + a3b3a1d(

1

u1 1

u3v2, u3, u1)

= a1b2a3d(1

u3 1

u1u1, v2, u3+ 1

u3u1u3, u1, v2+

1

u1 1

u3v2, u3, u1)

= a1b2a3d(1

u3u1u1, v2, u3 −

1

u3u21u1, u3u1, v2+

1

u1u3v2, u3, u1+

− 1

u1u23v2, u3u3, u1+

1

u1u3u3, u1, v2 −

1

u1u23u3, u1u3, v2 −

1

u3u21v2, u3u3, u1)

= a1b2a3d(1

u3u1(u1, v2, u3+ v2, u3, u1+ u3, u1, v2)+

−u1, v21

u3u21(u3, u1+ u1, u3)− u3, u1

1

u1u23(u3, v2+ v2, u3))

♥ ♦♥t râ à ♥tté ♦ r♦t −,− q

(u1, v2, u3+ v2, u3, u1+ u3, u1, v2) = 0

♦ù

−u1, v21

u3u21(u3, u1+ u1, u3)− u3, u1

1

u1u23(u3, v2+ v2, u3 = 0

♦♥t ♣r été ré

s éts ♣r s♦♥t ♦♥♥és ♥ ♥♥①

♦♥t u1 ∈ S t a1, b2, b3, v2, v3 ∈ A − S. ♣rès é♥t♦♥ [−,−]ω t s

♣r♦♣rétés −,−, ♥♦s ♦♥s

[[u1du1u1

, b2dv2]ω, b3dv3]ω =a1u1

u1, b2v2, b3dv3 +b3a1u1

u1, b2, v3dv2

+b3u1

a1, v3u1, b2dv2 −b3a1u21

u1, b2u1, v3dv2 +a1b3u1

u1, b2d(v2, v3)

+b2u1

a1, v2u1, b3dv3 + b3b2a1, v2, v3du1u1

+ b3b2, v3a1, v2du1u1

+b2b3a1, v2d(1

u1u1, v3) +

a1b2u1

u1, v2, b3dv3 −a1b2u21

u1, v2u1, b3dv3+

b3a1b2, v3d(1

u1u1, v2) + b3b2a1, v3d(

1

u1u1, v2) + a1b2b3d(

1

u1u1, v2, v3),

[[b2dv2, b3dv3]ω, u1du1u1

]ω = b2v2, b3v3, a1du1u1

+a1b2u1

v2, b3, u1dv3

+a1u1

b2, u1v2, b3dv3 + a1b2v2, b3d(1

u1v3, u1) + b3b2, v3v2, a1

du1u1

+a1b3u1

b2, v3, u1dv2 +a1u1

b3, u1b2, v3dv2 + a1b3b2, v3d(1

u1v2, u1)

+b1b3v2, v3, a1du1u1

+a1b2u1

b3, u1d(v2, v3) +a1b3u1

b2, u1d(v2, v3)

+a1b2b3d(1

u1v2, v3, u1)

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

t

[[b3dv3, u1du1u1

]ω, b2dv2]ω =b3u1

v3, a1u1, b2dv2 + b2b3v3, a1, v2du1u1

+b2b3, v2v3, a1d(1

u1u1, v2) +

a1u1

b3, u1v3, b2dv2 +b2a1u1

b3, u1, v2dv3

+b2u1

a1, v2b3, u1dv3 −b2a1u21

u1, v2b3, u1dv3 +a1b2u1

b3, u1d(v3, v2)

+b3a1u1

v3, u1, b2dv2 −b3a1u21

u1, b2v3, u1dv2 + b3b2a1, v2d(1

u1v3, u1)+

+b2a1b3, v2d(1

u1v3, u1) + a1b2b3d(

1

u1v3, u1, v2)

♣rès rr♦♣♠♥t ♦♥ ♠♦♥tr à ♥tté ♦ r♦t P♦ss♦♥

−,− q

[[b3dv3, u1du1u1

]ω, b2dv2]ω + [[b2dv2, b3dv3]ω, u1du1u1

]ω + [[u1du1u1

, b2dv2]ω, b3dv3]ω = 0

♠ê♠ ♦♥ ♠♦♥tr q

♥ t♥t s ssttt♦♥s

a1 // a2 u1 // u2 b2 // b3 v2 // v3 b3 // b1 t

v3 // v1 ♦♥ ♦t♥t

[[b1dv1, u2du2u2

]ω, b3dv3]ω+[[b3dv3, b1dv1]ω, u2du2u2

]ω+[[u2du2u2

, b3dv3]ω, b1dv1]ω = 0

♥ t♥t s ssttt♦♥s a1 // a3 u1 // u3 b2 // b1

v2 // v1 b3 // b2 t v3 // v2 ♦♥ ♦t♥t

[[b2dv2, u3du3u3

]ω, b1dv1]ω+[[b1dv1, b2dv2]ω, u3du3u3

]ω+[[u3du3u3

, b1dv1]ω, b2dv2]ω = 0

è ♣r ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦♥t u1 ∈ S t a1, b2, b3, v2, v3 ∈ A−S. ♣rès é♥t♦♥

[−,−]ω t s ♣r♦♣rétés −,−, ♥♦s ♦♥s

[[b3dv3, u1du1u1

]ω, b2dv2]ω + [[b2dv2, b3dv3]ω, u1du1u1

]ω +

[[u1du1u1

, b2dv2]ω, b3dv3]ω = 0

[[b1dv1, u2du2u2

]ω, b3dv3]ω + [[b3dv3, b1dv1]ω, u2du2u2

]ω +

[[u2du2u2

, b3dv3]ω, b1dv1]ω = 0

[[b2dv2, u3du3u3

]ω, b1dv1]ω + [[b1dv1, b2dv2]ω, u3du3u3

]ω +

[[u3du3u3

, b1dv1]ω, b2dv2]ω = 0.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

é♦ s ♣r♦♣♦st♦♥s t ♥érté [−,−]ω,

q

é♦rè♠ ♦t (A, −,−) ♥ èr P♦ss♦♥ ♦rt♠q ♣r♥

♣ ♦♥ ♥ é I ♥♥ré ♣r S := u1, ..., up. r♦t [−,−]ω é♥ ♣r

♥t sr ΩA(log I) ♥ strtr èr

♦♥strt♦♥ ♥ r♣rés♥tt♦♥ ♣r s ért♦♥s ♦rt

♠qs ΩA(log I)

♦s s♦♥s q ♣♣t♦♥ ♠t♦♥♥♥ H : ΩA → DerA t♦t strtr

P♦ss♦♥ sr A st ♥ ♠♦r♣s♠ èrs stss♥t ♦♥t♦♥

♦♠♣tté r à s♦s st♦♥ ♥♦s ♠♦♥tr♦♥s q ♥s s s

strtrs P♦ss♦♥ ♦rt♠qs ♣r♥♣s ♦♥ I tt ♣♣t♦♥ s

♣r♦♦♥ ♥ ♥ ♣♣t♦♥ A♥ér H : ΩA(log I) → DerA(log I). ♦t x ①é

♥s ΩA(log I) ♣♣t♦♥ ρω(x) : A → A é♥ ♣r

ρω(x)(a) = ω(x, d(a)) st ♥ Rért♦♥ sr A.♥ t ♣♦r t♦t a ∈ A,ρω(x)(a) =

p∑i=1xiρω(

duiui

)(a) +n∑p+1

xiρω(dvi) =p∑i=1

xiuiui, a+

n∑p+1

xivi, a

♦♥ ρω(x) =p∑i=1

xiuiui,−+

n∑p+1

xivi,−. ♦♥ ρω(x) st ♥ ért♦♥ ♦rt

♠q ♦♠♠ s♦♠♠ s ért♦♥s ♦rt♠qs ♥s ①st ♥ ♦♠♦♠♦r

♣s♠ A♠♦ ρω : ΩA(log I) → DerA(log I) q à t♦t x ∈ ΩA(log I)ss♦ ρω(x).

♥ ρω = H.

♦♥t u ∈ I∗ t a, b ∈ A ts q adu

u∈ ΩA(log I). ♣rès q ♣réè

ω(adu

u, db) =

a

uu, b. ♦♥ ρω(a

du

u)(b) =

a

uu, b =

a

u(ad(u))(b) t ♦♥

ρω(adu

u) =

a

uu,−.

♥s

ρω[adu

u, bdv

v] = ρω

(a

uu, bdv

v+b

va, vdu

u+ abd(

1

uvu, v)

)

=a

uu, bρω(

dv

v) +

b

va, vρω(

du

u) + abρω(d(

1

uvu, v))

=a

uvu, bv,−+ b

vua, vu,−+ ab 1

uvu, v,−

=a

uvu, bv,−+ b

vua, vu,−+ ab

uvu, v,−+

− ab

u2vu, vu,− − ab

uv2u, vv,−

Pr rs ♦♥

♦♥strt♦♥ érq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

ρω(adu

u)

(ρω(b

dv

v)

)=

a

uu, b

vv,−

=a

uvu, bv,−+ ab

uu, 1

vv,−

=a

uvu, bv,−+ ab

uvu, v,− − ab

uv2u, vv,−,

ρω(bdv

v)

(ρω(a

du

u)

)=

b

vv, a

uu,−

=b

uvv, au,−+ ab

vv, 1

uu,−

=b

uvv, au,−+ ab

uvv, u,− − ab

vu2v, uu,−.

♦♥ [ρω(adu

u), ρω(b

dv

v)] =

a

uvu, bv,− +

ab

uvu, v,− − ab

uv2u, vv,− +

b

uvv, au,−− ab

uvv, u,−+ ab

vu2v, uu,−. r ♣rès ♥tté ♦

u, v,− − v, u,−+ −, u, v = 0.

♦♥ [ρω(adu

u), ρω(b

dv

v)] =

a

uvu, bv,− − ab

uv2u, vv,− − b

uvv, au,− +

ab

vu2v, uu,−+ ab

uvu, v,− − ab

uv2u, vv,−.

t ♦♥ ρω([adu

u, bdv

v]ω) = [ρω(a

du

u), ρω(b

dv

v)].

♠ê♠ ♦♥ é♠♦♥tr q ρω[adu

u, bdv] = [ρω(a

du

u), ρω(bdv)] t ρω[adu, bdv] =

[ρω(adu), ρω(bdv)]. ♥ ♦♥ ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦t (A; −′−) ♥ èr P♦ss♦♥ ♦rt♠q ♦♥

I. ♣♣t♦♥ A♥ér ρω : x 7→ ρω(x) st ♥ ♠♦r♣s♠ èrs

♠rq♦♥s ss q

[du

u, adv

v]ω =

1

uu, adv

v+ ad(

1

uvu, v)

=1

uu, adv

v+ a[

du

u,dv

v]

=

(1

uu,−

)(a)

dv

v+ a[

du

u,dv

v]

= ρω(du

u)(a)

dv

v+ a[

du

u,dv

v]

♣r♦♣♦st♦♥ s♥t é♥érs tt ♣r♦♣rété

Pr♦♣♦st♦♥ P♦r t♦s ωj ∈ ΩA(log I) t f ∈ A, ♦♥

[ωi, fωj ] = f [ωi, ωj ] + (ρω(ωi)(f))ωj .

Pr ❱♦r ♥♥①

♥ ♥s é♠♦♥tré é♦rè♠ s♥t

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

é♦rè♠ ♦t strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ ♥

é I ♥ Rèr A ♥t sr ΩA(log I) ♥ strtr ♥rt

tr♠♥t t ♣♦r t♦t strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ ♦♥ ♥

é I, (ΩA(log I), ρω, [−,−]) st ♥ èr ♥rt

②♥t ♠♥ sr ΩA(log I), ♥ strtr ♥rt ♦♥ ♣t ♦ré♥♥t

♣♣qr t♥q Ps t ♥rt ♣♦r ♦♥strr ♥ ♦♠♣① ♥

q ♥♦s ♣r♠ttr ♦♥strr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q P♦r

♣♦s♦♥s LaltA(ΩA(log I),M) ♥s♠ s ♣♣t♦♥s ♠t♥érs ♥t

s②♠étrqs sr ΩA(log I) à rs ♥s ♥ A♠♦ ♥rt M.

LaltA(ΩA(log I),M) st ♥ Rèr ré ♦♠♠tt ♣♦r ♦

sté♥ ♣r

α ∧ β(xi, ..., xp+q) =∑

σ

εσµ(α(xσ(1), ..., xσ(p))⊗ β(xσ(p+1), ..., xσ(p+q)))

♦ù µ : M ⊗M → M st ♥ ♠♦r♣s♠ ΩA(log I)♠♦s ♠♥ ér♥

t dρω rt♥②♥r ss♦é à r♣rés♥tt♦♥ ρω é♥ ♣r

dρω(f)(α0, ...αp) =p∑i=0

(−1)iρω(αi)f(α0, ...αi, ...αp)+∑i,j(−1)i+jf([αi, αj ], α0, ..., αi, ..., αj , ..., αp)

♦♥ ♦t♥t ♥ èr ér♥t ré ♦♥t ♦♦♠♦♦ ss♦é st

♥♦té H∗PS (A, −,−;M) . r H∗

PS (A, −,−;M) = Ext(U(A,ΩA(log I)))(A,M) s

ΩA(log I) ♦♠♠ A−♠♦ st ♣r♦t ♥ ♦♣t ♦♥ é♥t♦♥ s♥t

é♥t♦♥ H∗PS (A, −,−;M) st ♣♣é ♦♦♠♦♦ P♦ss♦♥ ♦rt

♠q à rs ♥s ♠♦ ♥rt M.

♣rès q ♣réè ♦r♠ ω ♥t ♣r tt strtr P♦ss♦♥ st ♥

éé♠♥t Lalt2A(ΩA(log I),A). ♥ s ♠♥ s ♥st ♣s ♥ ♦②

♦♠♣① Lalt∗A(ΩA(log I),A). ♣r♦♣♦st♦♥ s♥t ♣♣♦rt ♥ ré♣♦♥s à tt

♣ré♦♣t♦♥

Pr♦♣♦st♦♥ ♦r♠ ω st ♥ ♦② Lalt∗A(ΩA(log I),A).

Pr ❱♦r

♥ ♥♦tr [ω−,−] ss ♦♦♠♦♦ ω

é♥t♦♥ [ω−,−] st ♣♣é ss P♦ss♦♥ ♦rt♠q

(A, −,−; I).

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦

P♦ss♦♥ ♦rt♠q

♥ ♠♦♥tr ♥s ❬t♦ ❪ q ♣r (ΩX,p(logD), DerX,p(logD)) st ré①

P♦r t♦t δ ∈ DerX,p(logD) t α ∈ ΩX,p(logD) ♦♥ ♥♦t

(δ|α) = iδ(α).

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♦ù (−|−) és♥ r♦t té ♥tr ΩX,p(logD) t DerX,p(logD). ♦s

♦♥s ♠♦♥tré ♣tr q t♦ts s st♦♥s ΩX(logD) s♦♥t ♦r♠

gα =dh

h+ η

♦ù g st ♥ ♦♥t♦♥ ♦♦♠♦r♣ srD t q dimCD∩z ∈ U : g(z) = 0 ≤ n−2

r ♣rès s ♦♥strt♦♥s st♦♥ ♣♣t♦♥ ♠t♦♥♥♥ H ♥

s é♥t q ♣♦r s ♦r♠s ②♥t s ♣ôs ♥q♠♥t sr D. tr♠♥t t

s ♦♥strt♦♥s ♥ s♦♥t ts q ♣♦r ♥ ss ♣rés srs P♦r

ts srs ΩX,p(logD) ♦t êtr ♥♥ré ♣r s ♦r♠s r♠és s rtérs

tqs ♥ t sr s♦♥t ♦♥♥és ♣r é♦rè♠

♥s tt st♦♥ ♥♦s s♣♣♦sr♦♥s ♥ ♣s q D stst s ②♣♦tèss

é♦rè♠ t q ♦♥t♦♥ é♥t♦♥ h D st rrét Pr rs

♦♥ s♣♣♦s q X st éq♣é ♥ strtr P♦ss♦♥ −,− ♦rt♠q

♦♥ é ID é♥t♦♥ D. ♦♠♠ à st♦♥ ♥♦s és♥r♦♥s ♣r

H : ΩX → DerX ♣♣t♦♥ ♠t♦♥♥♥ ss♦é à tt strtr P♦ss♦♥

♦tr s♦ st ♣r♦♦♥r H sr ΩX(logD) Psq t♦ts s st♦♥s

ΩX(logD) ♣♥t s ♠ttr s♦s ♦r♠ ω = gdh

h+η

g g ∈ OX t η ∈ ΩX

♦rs ♣♦r t♦t δ ∈ DerX(logD), ω ∈ ΩX(logD) ♦♥

Lδgω = −1

g

δh

h

dh

h+

1

ghLδdh+

1

gLδη −

δ.g

gω

H ét♥t ♣♣t♦♥ ♠t♦♥♥♥ ♥ strtr P♦ss♦♥ ♦♥

P♦r t♦t α1, α2, α3 ∈ ΩX

(H(α1)|α2) + (α1|H(α2)) = 0 (s♦tr♦♣)

(LHα1α2|Hα3)+ = 0

♥ ♣♦s

Hα := H(α0dh

h+ α1) =

α0

hH(dh) +H(α1.)

st ♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é à strtr P♦ss♦♥

♦rt♠q ♣r♥♣ ♣♣t♦♥ ♠t♦♥♥♥ H. Pr rs

Lα0

hH(dh)+H(α1)

α =α0

hLH(dh)α−H(dh).α

dα0

h+α0

hH(dh)α

dh

h+ LH(α1)α

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♥ s ♠♥r s H ér s rt♦♥s s♠rs à P♦r ♦♥ r♠r

qr q ♣♦r t♦t αi =dh

h+ α1

i , i = 1, 2, 3 ♦♥

(LH(α1)

α2|H(α3))

=

(1

hLH(dh)α

12|1

hH(dh)

)−

(H(dh)(α1

2)

h

dh

h| 1hH(dh)

)+

(1

hLH(dh)α

12|H(α1

3)

)

−(H(dh)(α1

2)

h

dh

h|H(α1

3)

)−(H(α1

1)h

h

dh

h| 1hH(dh)

)−

(H((α1

1)

h

dh

h|H(α1

3)

)

+

(1

hLH(α1

1)dh|

1

hH(dh)

)

+

(1

hLH(α1

1)dh|H(α1

3)

)+

(LH(α1

1)α

12|1

hH(dh)

)+

(LH(α1

1)α

12|H(α1

3))

♥ ♣♦s

Gr(H) := H(α)⊕ α, α ∈ Ω1X(log ID)

s étés t ♥♦s ♣r♠tt♥t é♠♦♥trr té♦rè♠ s♥t ♦♥t

s éts ♣r s♦♥t r♥♦②és ♥ ♥♥①

é♦rè♠ ♦t H ♣♣t♦♥ ♠t♦♥♥♥ ♥ strtr P♦ss♦♥

♦rt♠q ♣r♥♣ ♣♣t♦♥ ♠t♦♥♥♥ H.

H : ΩX(logD) → DerX(logD) stst s ♣r♦♣rétés s♥ts

Gr(H) st s♦tr♦♣

P♦r t♦t αi, αj , αk ∈ ΩX(log ID) ♦♥ (LH(αi)

α2|H(α3))+ = 0

♥ ♥ ét ♦r♦r s♥t

♦r♦r P♦r t♦ts st♦♥s α1, α2 ΩX(logD), ♦♥

[Hα1, Hα2] = H(iHα1dα2 − iHα2

dα1 + d(Hα1, α2))

Pr ♦♥t α1, α2 ① st♦♥s ΩX(logD), ♣rès té♦rè♠ ♦♥

0

= −(LHα1α2|Hα3)+

= (HLHα1α2|α3)− (LHα2

α3, Hα1)− (LHα3α1|, Hα2)

= (HLHα1α2|α3)− (LHα2

α3, Hα1)− (iHα3dα1 + diHα3

α1|Hα2)

= (HLHα1α2|α3)− (LHα2

α3|Hα1)− dα1(Hα3|Hα2)− (diHα3α1|Hα2)

= (HLHα1α2|α3)− (LHα2

α3|Hα1) + dα1(Hα3|Hα2)− (diHα3α1|Hα2)

= (HLHα1α2|α3)− (LHα2

α3|Hα1) + (iHα2dα1|Hα3)− (diHα3

α1|Hα2)

= (HLHα1α2|α3)− (HiHα2

dα1, α3)− (diHα3α1|Hα2)− (LHα2

α3|Hα1)

= (HLHα1α2 − HiHα2

dα1|α3)− (diHα3α1|Hα2)− (LHα2

α3, Hα1)

= (HLHα1α2 − HiHα2

dα1|α3)− (diHα3α1|Hα2)− (iHα2

dα3 + diHα2α3|Hα1)

= (HLHα1α2 − HiHα2

dα1|α3)− (d(iHα3α1)|Hα2)− (iHα2

dα3|Hα1)− (diHα2α3|Hα1)

= (HLHα1α2 − HiHα2

dα1, α3)− Hα2(Hα3|α1)− Hα1(Hα2|α3)− dα3(Hα2, α1)

= (HLHα1α2 − HiHα2

dα1|α3)− Hα2(Hα3|α1)− Hα1(Hα2|α3) + (α3, [Hα2, Hα1])+

+Hα1(α3|Hα2)− Hα2(α3|Hα1)

= (HLHα1α2 − HiHα2

dα1 − [Hα2, Hα1]|α3)

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

♦ù réstt

qs strtrs èr ss♦és ① strtrs P♦ss♦♥ ♦rt♠qs

P♦r t♦t m ∈ MD := OX [D] ♦♥ Hd log(m) ∈ DerX(logD).

MD és♥ s♦ss MX s s ♦♥t♦♥s ♠ér♦♠♦r♣s sr

X s ♦♥t♦♥s ♠ér♦♠♦r♣s ♣ôs sr D.

♦s r♣♣♦♥s q ♣♦r t♦t strtr P♦ss♦♥ ♦♦♠♦r♣ é♥ ♣r ♥

♣♣t♦♥ ♠t♦♥♥♥ H, s ♦♥t♦♥s ♠t♦♥♥♥s s♦♥t t♦ts ♦♦♠♦r♣s

sr X. ♥ ét rt♦♥ Hd log(m) ∈ DerX(logD) q rt♥s sss

♦♥t♦♥s ♠ér♦♠♦r♣s ♣♥t êtr ♠t♦♥♥♥s ♦rt♠qs st s s

♦♥t♦♥s ♠ér♦♠♦r♣s t②♣

m =

r∏

i=1

g

hrii

♦ù h =r∏i=1hi st ♦♥t♦♥ é♥t♦♥ D t g ♥ ♦♥t♦♥ s♥s ③ér♦s sr D.

♦s r♣♣♦♥s q t♦t strtr P♦ss♦♥ −,− s ♣r♦♦♥ ♠♥èr ♥q

♣r

u, abs =

1

bu, a − a

b2u, b

♥ ♥ ♥q strtr P♦ss♦♥ −,−s sr MX .

♥ st q ♣♦r t♦t m1,m2 ∈ MD s ①st♥t λ1, λ2 ∈ Z t a1, a2 ∈ O∗X ts q

mi =aihλi

tdmi

mi= ε(λi)λi

dh

h+daiai

♦ù ε(λi) és♥ s♥ λi.

♥ ♦♥sèr sr MD r♦t s♥t

m1,m2D =

(Hdm1

m1|dm2

m2) s mi ∈ MD −OX

(Hdm1|dm2

m2) s m2 ∈ MD −OX t m1 ∈ OX

(Hdm1|dm2) s mi ∈ OX

r♦t ♣♦ssè s ♣r♦♣rétés s♥ts

Pr♦♣♦st♦♥ r♦t −,−D ér s étés s♥ts

−,−D st C♥ér ♥ts②♠étrq

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

m1,m2D =

1

m1m2m1,m2s s mi ∈ MD −OX

1

m2m1,m2s s m2 ∈ MD −OX t m1 ∈ OX

m1,m2 s mi ∈ OX

,

−,−D st ♥ ért♦♥ ♦rt♠q sr MD − OX ♥ ♥ ss

♦♠♣♦s♥ts

P♦r t♦s m1,m2 ∈ MD −OX ,1

m1m2m1,m2s ∈ OX .

Pr é♦ s ♣r♦♣rétés ♦r♠s ♦rt♠qs

♥ ♥ ét ♦r♦r s♥t

♦r♦r −,−D st ♥ strtr sr s MD ♣r♦♦♥♥t

−,−Pr t ♠♦♥trr q −,−D ér ♥tté ♦ sr s st♦♥s

rst♥ts MD. P♦r ♦♥ st♥r tr♦s s

s u, v ∈ MD −OX t a ∈ OX .

s v ∈ MD −OX t a, b ∈ OX .

s u, v, w ∈ MD −OX .

♥ s♣♣♦s u, v ∈ MD −OX t a ∈ OX ♦rs

u, v, aDD = u, 1vv, asD

=1

uvu, v, ass −

1

uv2u, vsv, as

s♥st ♦♥ q

u, v, aDD+ =

=1

uvu, v, ass −

1

uv2u, vsv, as+

1

uvv, a, uss −

1

u2va, usv, us+

1

uva, u, vss −

1

uv2u, vsa, vs

− 1

u2vu, vsa, us

ç♦♥ ♥♦ ♦♥ ♠♦♥tr s trs s s trs éts tt ♣r s♦♥t

♦♥♥és ♥ ♥♥①

♥ ♥ ér s ♣r♦♣rétés s♥ts s r♦ts −,−s t −,−DPr♦♣♦st♦♥ s r♦ts −,−s t −,−D ér♥t s ♣r♦♣rétés s

♥ts sr MX −OX

m1,m2D(m1,m2D,m3D + m2,m3D + m1,m3D)+ = 0

1

m1m2,m3sm2,m3s,m1s−

1

m2m3m2,m1s−

1

m3m1m3,m1s+ = 0

♦♥strt♦♥ é♦♠étrq ♦♦♠♦♦ P♦ss♦♥

♦rt♠q

trtrs èr ♥rt sr ΩX(logD)

Préé♠♠♥t ♥♦s ♦♥s ♠♦♥trr q t♦t strtr P♦ss♦♥ ♦rt♠q

♥t ♥ ♠♦r♣s♠ s① H : ΩX(logD) −→ DerX(logD). ♦s ♦♥s à

♣rés♥t ♠♦♥trr q ♠♦r♣s♠ st ♥ strtr ♥rt

P♦r t♦t α := α1dh

h+ αi1dxi, β = β1

dh

h+ βj1dxj .

♥ é♥t sr ΩX(logD) r♦t s♥t

[α, β]

=α1

hh, β1

dh

h+β1hα1, h

dh

h+α1

hh, βjdxj+

+βjα1, xjdh

h+ α1β

jd(1

hh, βj) + αixi, β1

dh

h+

+β1hαi, hdxi + αiβ1d(

1

hxi, h) + αixi, βjdxj + βjαi, xjdxi + αiβjdxi, xj,

t ♦♥ été s♥t

[α, aβ] =

a(α1

hh, β1

dh

h+β1hα1, h

dh

h+α1

hh, βjdxj + βjα1, xj

dh

h+

+α1βjd(

1

hh, xj) + αixi, β1

dh

h+β1hαi, hdxi

+β1αid(

1

hxi, h) + αixi, βjdxj + βjαi, xjdxi + αiβjdxi, xj)

+α1

hh, adh

h+α1β

j

hh, adxj

+αiβ1xi, adh

h+ αiβjxi, adxj

q ♣rès rr♦♣♠♥t ♥♦s ♦♥♥

[α, aβ] = H(α)(a)β + a[α, β]

♥s ♦♣tq ♠♥r ΩX(logD) ♥ strtr èr ♥rt ♠♦♥

tr♦♥s ♠♠ ss♥t s♥t

♠♠ r♦t [−,−] é♥t ♥s ΩX(logD) ♥ strtr èr

♥rt

Pr ♦♥t α1, α2, α3 ∈ ΩX(logD).

[[α1, α2], α3]

= [LHα1α2 − iHα2

dα1, α3]

= −[α3,LHα1α2 − iHα2

dα1]

= iH(LHα1α2−iHα2

dα1)dα3 − LHα3

(LHα1α2 − iHα2

dα1).

r ♣rès ♦r♦r ♦♥

H(LHα1α2 − iHα2

dα1) = [Hα1, Hα2].

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦♥

[[α1, α2], α3]

= i[Hα1,Hα2]dα3 − LHα3

LHα1α2 + LHα3

iHα2dα1

= i[Hα1,Hα2]dα3 − LHα3

LHα1α2 + LHα3

LHα2− LHα3

diHα2α1

= i[Hα1,Hα2]dα3 − LHα3

LHα1α2 + LHα3

LHα2− dLHα3

iHα2α1.

.

Pr rs L ét♥t ♥ ♥♦♠♦r♣s♠ DerX(logD), s♥st q

[[α1, α2], α3]+ = i[Hα1,Hα2]dα3 + L[Hα3,Hα2]α1

− dLHα2iHα2

α1+

tr ♣rt ♦r♠ ♠q rt♥ t s étés ♣♣qés à α3 t

H(α2) ♦♥♥♥t

LHα2iHα2

α1 = i[Hα3,Hα2]+ iHα2

LHα3α1. tt r♥èr rt♦♥ ♥t q

−dLHα3iHα2

α1 = −d(LHα3

|Hα2

)− L[Hα3,Hα2]

α1 + i[Hα3,Hα2]dα1.

♥ sstt♥t −dLHα3iHα2

α1 ♥s ①♣rss♦♥ [[α1, α2], α3]+ sss ♦♥

♦t♥t

[[α1, α2], α3]+

= i[Hα1,Hα2]dα3 + L[Hα3,Hα2]

α1 − d(LHα3

|Hα2

)− L[Hα3,Hα2]

α1 + i[Hα3,Hα2]dα1+

= i[Hα1,Hα2]dα3 + i[Hα3,Hα2]

dα1 − d(LHα3

|Hα2

)+

= i[Hα1,Hα2]dα3 + i[Hα3,Hα2]

dα1 + i[Hα2,Hα3]dα1 + i[Hα1,Hα3]

dα2+

+i[Hα3,Hα1]dα2 + i[Hα2,Hα1]

dα3 − d((

LHα3|Hα2

)+

)

= d((

LHα3|Hα2

)+

)

r ♣rès é♦rè♠ (LHα3

|Hα2

)+ = 0.

♦ù réstt

♥ ♥ ét ss ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ H st ♥ strtr èr ♥rt sr ΩX(logD)

Pr ❱♦r ♥♥①

♥♦s ♣♦s♦♥s Laltp(ΩX(logD)) s s p♦r♠s OX ♥érs ♥t

s②♠étrqs sr ΩX(logD) t

Lalt(ΩX(logD)) =n⊕p=0

Laltp(ΩX(logD)). ♦rs ♣♣t♦♥

(∂Df)(α1, ..., αp) =n∑i=1

(−1)i−1H(αi)f(α1, ..., αi, ..., αp)+∑i≤j

(−1)i+jf([αi, αj ], α1, ..., αi, ..., αj , ..., αp)

ér

♠♠ ∂2D = 0

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

Pr é♦ rt♦♥ t ♥tté ♦ r♦t

[−,−].

♥ rést q (Lalt∗(logD), ∂D) st ♥ ♦♠♣① ♥s ♥ ♥ ét

é♥t♦♥ s♥t

é♥t♦♥ ♦♦♠♦♦ ♦♠♣① (Lalt∗(logD), ∂D) st ♣♣é ♦♦

♠♦♦ P♦ss♦♥ ♦rt♠q rété P♦ss♦♥ ♦rt♠q X

kime r♦♣ ♦♦♠♦♦ ♦♠♣① sr ♥♦té HkPS(X) t kime r♦♣

♦♦♠♦♦ P♦ss♦♥ ss♦é sr ♥♦té HkP (X).

①♠♣s s r♦♣s ♦♦♠♦♦s

P♦ss♦♥

♦rt♠qs

r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠qs s strtrs ♦s②♠♣tqs

♦t (L, [−,−], ρ, I) ♥ èr ♥rt ♦rt♠q ♦♠♠ ♥s

❬♦♥♦ ❪ ♥♦s ♦♣t♦♥s é♥t♦♥ s♥t

é♥t♦♥ ♥ ♣♣ strtr èr ♥rt P♦ss♦♥ ♥s L ♦

rt♠q ♦♥ I t♦t ♦r♠ µ sr L dρr♠é

♦rsq µ st ♥ strtr èr ♥rt P♦ss♦♥ ♦rt♠q

(L, ρ, µ) st ♣♣é èr ♥rt P♦ss♦♥ ♦rt♠q ♥ r q

L st ♥ èr ♥rt P♦ss♦♥ ♦rt♠q ♦rsq ①st sr L ♥

strtr ♥rt P♦ss♦♥ sr L ♦rt♠q A st ♥ èr P♦ss♦♥

♦r♠ ss♦é ω, ♦rs

dH(ω) = 0

♥ t ♣♦r t♦t a, b, c ∈ A ♦♥

dH(ω)(da, db, dc) = H(da)ω(db, dc)−H(db)ω(da, dc) +H(dc)ω(da, db)

−ω(da, b, dc) + ω(da, c, db)− ω(db, c, da)= −2(Jacobi(a, b, c))

♥ ♦♥t ♦♥ q t♦t ♦r♠ P♦ss♦♥ sr A ♥t sr ΩA ♥ strtr

èr ♥rt P♦ss♦♥ ♥ st ♠ê♠ ♣♦r s ♦r♠s P♦ss♦♥

♦rt♠qs q ♥s♥t sr Ω1A(log I) s strtrs èr ♥rt

P♦ss♦♥ ♦rt♠qs

tt ♥♦t♦♥ strtr èr ♥rt P♦ss♦♥ ♦rt♠q st é

strtr èr ♥rtP♦ss♦♥s②♠♣tq ♦rt♠q

♥ P t réér♥ à ♥s P♦ss♦♥ ♦rs st ♠s ♣♦r ②♦ t♦

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

é♥t♦♥ ♥ ♣♣ strtr èr ♥rtP♦ss♦♥s②♠♣tq

♦rt♠q ♥s L t♦t strtr èr ♥rt P♦ss♦♥

♦rt♠q µ sr L ♣♦r q ♣♣t♦♥

L → HomA(L,A)

x 7→ ıxµ

st ♥ s♦♠♦r♣s♠ A♠♦s

s♥ st q ♥s t♦t rété ♦s②♠♣tq (X,ω,D) ω st ♥ strtr

èr ♥rtP♦ss♦♥ s②♠♣tq ♦rt♠q sr DerX(logD).

♦s ♦♥s à ♣rés♥t ♠♦♥trr q s strtrs èr ♥rtP♦ss♦♥

s②♠♣tq ♦rt♠q s♦♥t ♥tèr♠♥t rtérsés ♣r H.

♦t µ ♦r♠ ss♦é à ♥ strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ −,− ♦♥ ♥ é I A t µ ♥ strtr èr ♥rtP♦ss♦♥

s②♠♣tq ♦rt♠q

♥s st s strtrs èr ♥rts②♠♣tq ♦rt♠qs

sr♦♥t s♠♣♠♥t ♣♣é strtr èr ♥rt♦s②♠♣tq

Pr♦♣♦st♦♥ µ st ♥ strtr èr ♥rt♦s②♠♣tq

sr ΩA(log I) s t s♠♥t s H st ♥ s♦♠♦r♣s♠ A♠♦s

Pr ♣♣♦s♦♥s q H st ♥ s♦♠♦r♣s♠

♦t x, y ∈ ΩA(log I) ts q I(x) = I(y).

♦rs

−σ(H(x)) = I(x)

= I(y)

= −σ(H(y))

i.e. H(x) = H(y)

i.e. x = y

é♣r♦q♠♥t s♦t ψ ∈ H(ΩA(log I),A). r♦♥s x ∈ ΩA(log I) t q I(x) =ψ.

Psq ψ ∈ Hom(ΩA(log I),A) ∼= DerA(log I) = H(ΩA(log I)) ♦rs ①st

z ∈ ΩA(log I) t q H(z) = σ−1(ψ). ♥ ♦♥

I(−z) = σ(H(z)) = ψ.

st ♣r♥r x = −z.é♣r♦q♠♥t ♦♥ s♣♣♦s q I st ♥ s♦♠♦r♣s♠ ♦rs s H(x) = H(y),

♦rs −H(H(x)) = −H(H(y)); I(x) = I(y) t ♣r s♥t x = y.

Pr rs ♣♦r t♦t δ ∈ DerA(log I), ①st x ∈ ΩA(log I) t q

σ(δ) = I(x) = −σ(H(x)); H(−x) = δ.

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q x, y = x

sr C[x, y]

♥t ♥r tt ♦♦♠♦♦ rér♦♥s qqs réstts

té♦rqs ♥tr♦ts ♥ st♦♥ ♦t ♦♥ tt st♦♥ A és♥r èr

C[x, y]. ♥ ♦♥sèr sr A r♦t

(f, g) 7→ f, g = x(∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x)

q t A ♥ èr P♦ss♦♥ Pr rs ♣♦r t♦t f ∈ A, ért♦♥

Df := x(∂f

∂x

∂

∂y− ∂f

∂y

∂

∂x)

ér

Df (xA) ⊂ xA

st ♦♥ ♥ ért♦♥ ♦rt♠q ♣r♥♣ ♦♥ I = xA. ♥ ♥

ét ♦♥ q A st ♥ èr P♦ss♦♥ ♦rt♠q ♦♥ é xA.♦♠♠ t♦t strtr P♦ss♦♥ ♥t ♥ ♣♣t♦♥ A♥ér H : ΩA →DerA é♥ ♣r

H(df) = Df

♣♥♥t éqt♦♥ ♥♦s és♦♥s q H(ΩA) ⊂ Der(log xA) ♦ù

Der(log xA) st As♦s ♠♦ DerA ♦r♠é s ért♦♥s ♦rt♠qs

♦♥ I.♥ ♦♥

H(dx) = Dx = x∂

∂y, H(dy) = Dy = −x ∂

∂x

♦♥ r♠rq ss q

1

xDx(xA) =

∂

∂y(xA) = x

∂

∂y(A) ⊂ xA.

s♥st ♦♥ q1

xDx(xA) ∈ Der(log xA).

♦♥ ♦♥

H(dx

x) =

1

xH(dx) =

∂

∂yt H(dy) = H(dy) = −x ∂

∂x

♠♠ s♥t ♥♦s ♣r♠t ♦♥r q s ♦♥♥és ss♥t ♣♦r é♥r

♥tèr♠♥t H.

♠♠

ΩA(log I) ∼= Adxx

⊕Ady ∼= C[y]dx

x⊕ ΩA.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♥t ♦♥ q ♣♦r t♦t α ∈ ΩA(log I), ①st a, b ∈ A ts q

α = adx

x+ bdy.

♥ ♦t♥t ♦♥

H(adx

x+ bdy) = −bx ∂

∂x+ a

∂

∂y∈ Der(log xA).

♥ é♥t ♥s ΩA(log I) r♦t

[α01

dx

x+ α1

1dy, α02

dx

x+ α1

2dy] :=(α01

xx, α0

2+α02

xα0

1, x+ α12α0

1, y+ α11y, α0

2)dx

x+

(α01

xx, α1

2+α02

xα1

1, x+ α11y, α1

2+ α12α1

1, y)dy .

♥ ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ r♦t é♥ ♣r ♥t ♥ strtr èr

sr ΩA(log I).

Pr ❱♦r ♥♥①

♠rq

P♦r t♦t a(y)dx

x∈ C[y]

dx

xt bdx+ cdy ∈ ΩA, ♦♥

[a(y)dx

x, bdx+ cdy] = [a(y)

dx

x, bdx] + [a(y)

dx

x, cdy]

= a(y)(∂b

∂y− b

∂a(y)

∂y)dx+ a(y)

∂c

∂ydy ∈ ΩA

♥ ♦♥t q ΩA st st ♣♦r r♦t ΩA(log I). Pr rs

[a(y)dx

x, b(y)

dx

x] = (a(y)

∂b(y)

∂y− b(y)

∂a(y)

∂y)dx

x

t

[[a(y)dx

x, b(y)

dx

x], c(y)

dx

x]+ =

[(a(y)∂b(y)

∂y− b(y)

∂a(y)

∂y)dx

x, c(y)

dx

x]+ =

(a(y)(∂b(y)

∂y− b

∂a(y)

∂y)∂c(y)

∂y− ca

∂2b(y)

∂yy+ cb

∂2a(y)

∂yy)dx

x+

= 0

♦♥ C[y]dx

xst st ♣♦r r♦t [−,−].

♥s s ♣rtr ♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é ér

♣r♦♣rété s♥t

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

♠♠ P♦r t♦t α = α01

dx

x+α1

1dy, β = β01dx

x+β11dy ∈ ΩA(log I) t a ∈ A,

♦♥

[α, aβ] = H(α)(a)β + a[α, β]

Pr ❱♦r ♥♥①

Pr♦♣♦st♦♥ H : ΩA(log I) −→ DerA(log xA) st ♥ ♦♠♦♠♦r♣s♠

èrs

Pr ❱♦r ♥♥①

P♦r ér s ♥♦tt♦♥s ♥♦s ♦♥sér♦♥s s s♦♠♦r♣s♠s s♥ts

Lalt0(ΩA(log I), I) ∼= A, Lalt1(ΩA(log I), I) ∼= DerA(log I) ∼= A × A t

Lalt2(ΩA(log I), I) ∼= A. s r♥rs ♦♠♣① P♦ss♦♥ ♦

rt♠q ss♦é ♥t

0 // Ad0H // A×A

d1H // A // 0

ou d0H(f) = (∂yf,−x∂xf) t d1H(f1, f2) = ∂yf2 + x∂xf1.

♥ ♥

d1H(d0Hf) = x(∂2xyf − ∂2xyf) = 0.

q ♠♦♥tr q dH st rré ♥

Pr♦♣♦st♦♥ ♦r♠ P♦ss♦♥ ss♦é à x, y = x st ♦s②♠♣

tq

Pr

Pr é♥t♦♥ ♦r♠ P♦ss♦♥ −,− st µ = x∂x ∧ ∂y. ♥ ♥ ét q

♦r♠ ♦rrs♣♦♥♥t à µ st ω =dx

x∧ dy q st ♥ ♦s②♠♣tq

♦s ♦♥s à ♣rés♥t r s r♦♣s ♦♦♠♦♦ ss♦és

Pr♦♣♦st♦♥ H0PS

∼= C H1PS

∼= C H2PS

∼= 0A.

Pr ♣rès é♥t♦♥ ér♥t dH , ♦♥

H0PS .

P♦r f ∈ A. f ∈ ker d0H

s t s♠♥t s∂f

∂y=∂f

∂x= 0.

s♥st q Kerd0H

∼= C.

H2PS .

P♦r t♦t g ∈ A ♦♥ g = d1H(0,

∫gdy+ k(x)). ♦♥ d1

Hst ♥ é♣♠♦r♣s♠

t ♣r st H2PS

∼= OA.

H1PS . ♥ r♠rq q A2 ∼= (C[y]× C[x])⊕ (xA× yA).

♥s ♣♦r t♦t (f1, f2) ∈ A × A, ①st g1 ∈ C[y], g2 ∈ C[x], h2, h1 ∈ At q f1 = g1(y) + xh1 t f2 = g2(x) + yh2. s ♣♦r t♦t (a(y), b(x)) ∈

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

C[y]×C[x], x∂a(y)

∂x+∂b(x)

∂y= 0. ♦♥ C[y]×C[x] ⊂ ker d1

H. P♦r s rs♦♥s

♥♦s ♥♦s ♦♥s

ker(d1H) : = ker(d1

H) ∩ A2

= (C[y]× C[x])⊕ ker(d1H) ∩ (xA× yA)

= (C[y]× C[x])⊕Θ(A)

♦ù Θ st é♥ ♣r

A Θ // A2 a 7→ (xa,−∫x∂xa

∂xdy)

Pr rs Θ(A) ⊂ ker(d1H) t

A ∼= C[x]⊕ yC[y]⊕ xyA.

♥ ♥ ét q ♣♦r t♦t f ∈ A, ①st (f1, q, p) ∈ C[x] × C[y] × A t

q f = f1 + yq + xyp.

♥s∂f

∂y= q+y

∂q

∂y+x(p+y

∂p

∂y) = (1+y

∂

∂y)q+x(1+y

∂

∂y)p ∈ C[y]⊕x(1+y ∂

∂y)(A)

t

−x∂f∂x

= −x∂f1∂x

− xyp− x2y∂p

∂x= −x∂f1

∂x− xy(1 + x

∂

∂x)p ∈ xC[x]⊕ xy(1 +

x∂

∂x)A.

♥ ♦♥sèr Ψ : A → A2, f 7→ (x(1 + y∂

∂y)f,−xy(1 + x

∂

∂x)f)

Psq

(x(1 + y∂

∂y)f,−xy(1 + x

∂

∂x)f) = (xf

∂y

∂y+ xy

∂f

∂y,−x∂x

∂xyf − x2

∂yf

∂x)

= (∂xyf

∂y,−x∂xyf

∂x)

= d0H(xyf)

t Ψ(A) ⊂ d0H(A). ♥

(∂f

∂y,−x∂f

∂x) ∈ (C[y]× xC[x])⊕Ψ(A)

é♣r♦q♠♥t ♣♦r t♦t F := (f1(y), xf2(x)) + Ψ(p) ∈ (C[y] × xC[x]) ⊕Ψ(A), ♥

F = d0H(

∫f1dy−

∫f2dx)+d

0H(xyp) = d0

H(

∫f1dy−

∫f2dx+xyp) ∈ d0

H(A)

♦♥

d0H(A) ∼= (C[y]× xC[x])⊕Ψ(A)

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

tr ♣rt t q d0H(∫xady) = (xa,−

∫x∂xa

∂xdy) ♣♦r t♦t a ∈ A,

♦♥ ♥ ét q Θ(A) ⊂ d0H(A). ♣s ♣r ♥ rt ♥♦s ♦♥♥

Θ(A) ⊂ Ψ(A).

Psq (C[y]×C[x]) ∼= (C[y]×xC)⊕ (0A×C) t x∂A∂x

∩C = 0A, ♦♥ ♦♥

d0H(A) ∩ (0A × C) ∼= 0A.

♦ù

H1PS

∼= C.

♣rès s Pr♦♣♦st♦♥s t s ♦♦♠♦♦s P♦ss♦♥ ♠

♦rt♠qs t P♦ss♦♥ ♦rt♠qs ss♦és à strtr P♦ss♦♥

x, y = x s♦♥t s♦♠♦r♣s ♣r♦♣♦st♦♥ s♥t ♥♦s ♣r♠t érr

réstt ♥s s strtr P♦ss♦♥ ♦rt♠q ♣r♥♣ x, y = x.

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦s P♦ss♦♥ x, y = x s♦♥t

H0P∼= C H1

P∼= C t H2

P∼= 0A.

Pr rs ♦♠♣① ♠ ♦rt♠q ♦♥ I st

0 // A d0 // Ω1A(log xA)

d1 // Ω2A(log xA) // 0

♦ù

d0(a) := x∂x(a)dx

x+ ∂y(a)dy

t

d1(adx

x+ bdy) := (x∂x(b)− ∂y(a))

dx

x∧ dy.

Pr♦♣♦st♦♥ r♠♠ s♥t st ♦♠♠tt

0 // A

d0 // ΩA(log xA)

−H

d1 // Ω2A(log xA)

−H

// 0

0 // Ad0H // A2

d1H // A // 0

Pr P♦r t♦t a ∈ A ♥♦s ♦♥s H(da) = H(x∂x(a)dx

x+ ∂y(a)dy) =

−∂y(a)x∂x + x∂x(a)∂y ∼= (−∂y(a), x∂x(a)) t d0H(a) ∼= (∂y(a),−x∂x(a)) = −H(da)

♣s ♣♦r t♦t α = fdx

x+gdy ∈ ΩA(log I), ♦♥ d1(α) = (x∂x(g)−∂y(f))

dx

x∧

dy, −H(d1(α)) ∼= x∂x(g)− ∂y(f).

Pr rs −H(α) = gx∂x − f∂y ∼= (g,−f) ♥♦s ♦♥s d1H(−H) = d1

H(gx∂x −

f∂y) ∼= x∂x(g)− ∂y(f)

♣r♦♣♦st♦♥ s♥t ♦♥♥ s r♦♣s ♦♦♠♦♦ ss♦és à ♦♠♣①

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦ ♦♠♣① s♦♥t H0DS

∼=C H1

DS∼= C t H2

DS∼= 0A.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

Pr

P♦r s♠♣r s ♥♦tt♦♥s ♥♦s ♣♦s♦♥s

Ω1A(log xA)

∼=→ A×Aadx

x+ bdy 7→ (a, b)

Ω2A(log xA)

∼=→ Aadx

x∧ dy 7→ a

s ♥♦tt♦♥s ♦♠♣① ♥t

0 // A d0 // A×A d1 // A // 0

♦ù d0(f) = (x∂xf, ∂yf) t d1(f1, f2) = x∂xf2 − ∂yf1.

P♦r t♦t f ∈ A, ♦♥ f = d1(−∫fdy, 0). ♦♥ A ∼= d1(A × A) t ♣r st

H2DS

∼= 0.

Pr ♥ s♠♣ ♦♥ ♦t♥t H0DS

∼= C.

♦t (f1, f2) ∈ A×A. (f1, f2) ∈ ker(d1) s t s♠♥t s f1 = x∫∂xf

2dy + k(x).

♦♥ ker(d1) ∼= (x∫∂xudy, u);u ∈ A ⊕ xC ⊕ C. r ♣♣t♦♥ s♥t st ♥

♠♦♥♦♠♦r♣s♠ ♠♦s

θ : A → xA×Au 7→ (x

∫∂xudy, u)

t ker(d1) ∼= θ(A)⊕ (xC× 0A) ∼= θ(A)⊕ (xC⊕ C).

♣s ♣♦r u ∈ A t a ∈ C[x], ♦♥

d0(∫udy+

∫adx) = (x

∫∂xudy+xa, u) = (x

∫∂xudy, u)+(xa, 0) = θ(u)+(xa, 0) ∈

θ(A)⊕ (xC). ♦♥ θ(A)⊕ (xC) ⊂ d0(A). Psq C ∩ d0(A) = 0A ♦♥ d0(A) =

d0(A)∩ (ker(d1)) ∼= θ(A)⊕ (xC). ♦♥ ker(d1) ∼= d0(A)⊕C. t ♦♥ H1DS

∼= C.

s ♦♦♠♦♦s P♦ss♦♥ t P♦ss♦♥ ♦rt♠q

(A := C[x, y], x, y = x2)).

♥s tt st♦♥ ♥♦s ♣r♦♣♦s♦♥s ♥ ①♠♣ strtr P♦ss♦♥ ♥♦♥

♦s②♠♣tq t ♥♦s ♠♦♥tr♦♥s q ss r♦♣s ♦♦♠♦♦s P♦ss♦♥ t

P♦ss♦♥ ♦rt♠q s♦♥t s♦♠♦r♣s

♥ ♦♥sèr ♥sA = C[x, y] r♦t P♦ss♦♥ x, y = x2 q st ♣r é♥t♦♥

♦rt♠q ♣r♥♣ ♦♥ é A = C[x, y] ♥♥ré ♣r x2. ♦t♦♥s

qdx2

x2= 2

dx

x. ♦♥ ΩA(log x

2A) st s♦♠♦r♣ A♠♦ ♥♥ré ♣r dxx

∪ΩA

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q A = C[x, y], x, y = x2.

♣♣t♦♥ ♠t♦♥♥♥ ♦rt♠q ss♦é st é♥ sr s é♥ér

trs ΩA(log x2A) ♣r H(

dx

x) = x∂y, H(dy) = −x2∂x. ♥ ♥ ét ♦♠♣①

P♦ss♦♥ ♦rt♠q s♥t

0 // Ad0H(H) // A×A

d1H // A // 0

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

♦ù d∗H

st é♥ ♣r d0H(f) = (x∂yf,−x2∂xf), d1H(f1, f2) = x∂yf2 + x2∂xf1 − xf1

t s s♦♠♦r♣s♠s s♥ts s♦♥t s♦s♥t♥s

DerA(log x2A)

∼=→ A×Aax∂x + b∂y 7→ (a, b)

DerA(log x2A) ∧DerA(log x2A)

∼=→ Aax∂x ∧ ∂y 7→ a

H2PS (A := C[x, y], x, y = x2).

Psq A ∼= C[y] ⊕ xA ♦rs ♣♦r t♦t g ∈ A, ①st g1, g2 ∈ A t q

g = g1 + xg2. ♥s ♣♦r t♦t g ∈ A ♦♥ g ∈ d1H(A) s t s♠♥t s g = xg2 =

x∂yf2 + x2∂xf1 − xf1. s xg2 = x∂y(x∫∂xg2dy) − x2∂xg2 − xg2 t éqt♦♥

x(∂yv + x∂xu − u) = g(y) ∈ C[y]∗ ♥ ♣♦ssè ♣s s♦t♦♥ ♥s A × A. ♦♥

A ∼= d1H(A×A)⊕ C[y]. s♥st q

H2PS

∼= C[y].

H1PS .

P♦r r H1PS ♥♦s ♦♥s s♦♥ ♠♠ s♥t

♠♠ ♦t ϕ : E → F ♥ ♠♦♥♦♠♦r♣s♠ s♣s t♦rs P♦r t♦t

s♦s ♥s♠ A,B E,ϕ(A⊕B) = ϕ(A)⊕ ϕ(B)

Pr st r q ϕ(A ⊕ B) = ϕ(A) + ϕ(B). z ∈ ϕ(A) ∩ ϕ(B), ♦rs

z ∈ ϕ(A⊕B) = 0E . Pr st ϕ(A⊕B) = ϕ(A)⊕ ϕ(B).

♦t (f1, f2) ∈ A×A.(f1, f2) ∈ ker(d1

H) s t s♠♥t s ①st k ∈ C[x] t q f2 =

∫(1−x∂x)f1dy+

k(x). ♥s ker(d1H) ∼= (u,

∫(1 − x∂x)udy), uA ⊕ C[x]. P♦r t♦t u ∈ A ♦♥ ♣♦s

η(u) = (u,∫(1− x∂x)udy). ♦♥ η : A → A×A st ♥ ♠♦♥♦♠♦r♣s♠ s♣s

t♦rs ker(d1H) ∼= η(A)⊕C[x] ∼= η(C[y])⊕η(xA)⊕C[x]; ♣sq A ∼= C[y]⊕xA.

tr ♣rt ♣♦r t♦t g ∈ η(xA) ⊕ (0A, x2C[x]), ①st u ∈ A t v ∈ C[x] ts

q g = (xu,−x2∫∂xdy + x2v(x)) = d0

H(∫udy −

∫v(x)dx). ♣s ♣♦r t♦t

u(y) ∈ C[y] t a0, a1 ∈ C, éqt♦♥ ér♥t

xfy = u(y)

−x2fx =∫u(y)dy + a0 + a1x

♥ ♣♦ssè ♣s s♦t♦♥ ♥s A. ♦♥ ker(d1H) ∼= η(C[y])⊕ C1[x]⊕ d0

H(A). Pr

st

H1PS

∼= η(C[y])⊕ C1[x].

♦ù C1[x] := a0 + a1x; a0, a1 ∈ C. tr ♣rt ♣sq η st ♥ ♠♦♥♦♠♦r♣s♠

η(C[y]) ∼= C[y]. ♦rs

H1PS

∼= C[y]⊕ C1[x].

è ♣r ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

x, y = x2 s♦♥t

H1PS

∼= C[y]⊕ C1[x];H2PS

∼= C[y], H0PS

∼= C.

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

♦♦♠♦♦ P♦ss♦♥ A = C[x, y], x, y = x2.

t♦♥ ♣♣t♦♥ ♠t♦♥♥♥ ss♦é sr s é♥értrs ΩA st

H(dx) = x2∂y t H(dy) = −x2∂x.♣r s♦ s♠♣té ♥♦s ♦♣t♦♥s s s♦♠♦r♣s♠s s♥t

DerA∼=→ A×A

a∂x + b∂y 7→ (a, b)

DerA ∧DerA∼=→ A

a∂x ∧ ∂y 7→ a.

s s♦♠♦r♣s♠s ér♥t ♦♦♠♦♦ P♦ss♦♥ ♥t

d0H(f) = (x2∂yf,−x2∂xf) t d1H(f1, f2) = x2∂xf1 + x2∂yf2 − 2xf1.

P♦r t♦t g ∈ A, ♥♦s ♦♥s xg = −2x(−1

2g) + x2(

1

2)(−∂xg + ∂y(

∫∂xgdy)).

♦♥ A ∼= d1H(A×A)⊕ C[y].

Pr st

H2P∼= C[y].

♦t (f1, f2) ∈ A×A

(f1, f2) ∈ ker(d1H) s t s♠♥t s u ∈ A, a ∈ C[x]. stàr f1 = xu t

f2 =∫(1− x∂x)udy + a(x).

♦♥ ker(d1H) = (xu,∫(1 − x∂x)udy + a(x)), u ∈ A, a(x) ∈ C[x]. ♥ ♣♦s

ϕ(u) = (xu,∫(1 − x∂x)udy ♣♦r t♦t u ∈ A. ♦rs ϕ : A → xA × A st ♥

s♦♠♦r♣s♠ s♣s t♦rs t

ker(d1H)∼= ϕ(A)⊕ C[x]

tr ♣rt ♦♠♣t t♥ t q A ∼= C[y]⊕xA, ♦♥ ϕ(A) ∼= ϕ(C[y])⊕ϕ(xA).

Pr rs ϕ(xA)⊕ x2C[x] ⊂ d0H(A), t ♦♥ d0H(A) ∩ ϕ(C[y])⊕ C1[x] ∼= 0APr st

ker(d1H)∼= ϕ(C[y])⊕ C1[x]⊕ d0H(A) ∼= C[y]⊕ C1[x]⊕ d0H(A)

♥ ♥ ét q

H1P∼= C[y]⊕ C1[x]

è ♣r ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ s r♦♣s ♦♦♠♦♦ P♦ss♦♥ x, y = x2 s♦♥t

H1P∼= C[y]⊕ C1[x];H

2P∼= C[y], H0

P∼= C

♥ ét é♥t♦♥ s ér♥ts ♦r♠s ♦rt♠qs ♦♥ x2At xA q ΩA(log x

2A) ∼= ΩA(log xA). ♥s ♦r♠ ω =dx

x2∧ dy ss♦é à

♦r♠ P♦ss♦♥ x2∂

∂x∧ ∂

∂y x, y = x2 ♥st ♣s ♦rt♠q t♥t

♣s q1

x/∈ C[x, y]. Pr rs ♦♠♦♠♦r♣s♠ ♠♦s s♥t

µ : ΩA(log x2A) → HA(ΩA(log x

2A),A), α0dx

x+ α0dy 7→ −α1x

2 ∂

∂x+ xα0

∂

∂y

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

♥st ♣s srt

♥ t −1

xdy st ♥q ♥téé♥t x

∂

∂xq ♥st ♣♥♥t ♣s éé♠♥t

ΩA(log x2A) s ♦♥ ♦♥sèr ♥♦tr é♥t♦♥ s ér♥ts ♦r♠s ♦rt

♠qs s♥st q x, y = x2 st ♥ strtr P♦ss♦♥ ♦rt♠q

♣r♥♣ ♥♦♥ ♦s②♠♣tq ♥ ♦t♥t ♦♥ té♦rè♠ s♥t

é♦rè♠ r♦t x, y = x2 ♥t sr C[x, y] ♥ strtr P♦s

s♦♥ ♦rt♠q ♣r♥♣ ♦♥ é x2A. tt strtr P♦ss♦♥ ♥st

♣s ♦s②♠♣tq ♠s ss r♦♣s ♦♦♠♦♦ P♦ss♦♥ t P♦ss♦♥ ♦

rt♠qs s♦♥t s♦♠♦r♣s

♦♦♠♦♦ P♦ss♦♥ t P♦ss♦♥ ♦rt♠q strtr P♦ss♦♥ x, y = 0, x, z =0, y, z = xyz sr A = C[x, y, z]

♥s tt ♣rt ♥♦s ♠♦♥tr♦♥s q strtr P♦ss♦♥ ♦rt♠q

♣r♥♣A = C[x, y, z] ♥st ♣s ♦s②♠♣tq t q ss r♦♣s ♦♦♠♦♦

t ① P♦ss♦♥ t P♦ss♦♥ ♦rt♠q ss♦és s♦♥t ér♥ts Pr é♥t♦♥

tt strtr P♦ss♦♥ st ♦rt♠q ♣r♥♣ ♦♥ é xyzA t s

ér♥ts P♦ss♦♥ ♦rt♠qs s♦♥t

d0H(f) = (0, xz

∂f

∂z,−xy∂f

∂y)

d1H(f1, f2, f3) = (xz

∂f3∂z

+ xy∂f2∂y

− xf1,−xy∂f1∂y

,−xz∂f1∂z

)

d2H(f1, f2, f3) = xz

∂f2∂z

+ xy∂f3∂y

.

♠ê♠ s ér♥ts P♦ss♦♥ ss♦és s♦♥t

δ0(f) = xyz(0,∂f

∂z,−∂f

∂y)

δ1(f1, f2, f3) = (xyz∂f3∂z

+ xyz∂f2∂y

− yzf1 − xzf2 − xyf3,−xyz∂f1∂y

,−xyz∂f1∂z

)

δ2(f1, f2, f3) = xyz(∂f2∂z

+∂f3∂y

).

H3PS

♦s és♦♥s s éqt♦♥s q d2H(A3) ⊂ xA.

♣♥♥t

A ∼= C[y]⊕ zC[z]⊕ xA∼= C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xzC[z]⊕ x2yA⊕ x2zA⊕ xyzA.

t ♣r ♦♥tr s♥r q ♣rès é♥t♦♥ ♦r♠ ér♥t ♦rt♠q

♦♥♥é ♥s ❬t♦ ❪ −dy

xst ♥ ♥ ♦r♠ ér♥t ♦rt♠q

♣tr ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

tr ♣rt ♣♦r t♦t xg(x) ∈ xC[x] éqt♦♥ ér♥t z∂u

∂z+y

∂v

∂y= g(x) ♥

♣♦ssè ♣s s♦t♦♥ ♥s A×A×A ♣s ♣♦r t♦t g ∈ xyC[y]⊕ xzC[z]⊕x2yA⊕ x2zA⊕ xyzA, ①st

g1(y), g2(z), g3(x, y, z), g4(x, y, z), g5(x, y, z) ∈ At q g = xyg1(y) + xzg2(z) + x2yg3(x, y, z) + x2zg4(x, y, z) + xyzg5(x, y, z)

♥ ♥ ét ①♣rss♦♥ s ♦♦rs

z∂f2∂z

+ y∂f3∂y

= yg1(y)+ zg2(z)+xyg3(x, y, z)+xzg4(x, y, z)+ yzg5(x, y, z)

q st éq♥t à

z(∂f2∂z

−g2(z)−xg4(x, y, z))+y(∂f3∂y

−g1(y)−xg3(x, y, z)−zg5(x, y, z)) = 0

st ♦♥ ♣r♥r

f2 =

∫g2(z) + xg4(x, y, z)dz; f3 =

∫g1(y) + xg3(x, y, z) + zg5(x, y, z)dy

♣♦r ♦r

d2H(A3) ∼= xyC[y]⊕ xzC[z]⊕ x2yA⊕ x2zA⊕ xyzA.

♥ ♥ ét q

H3PS

∼= C[y]⊕ zC[z]⊕ xC[x].

H3P .

éqt♦♥ ♥♦s és♦♥s q

δ2(A3) ⊂ xyzA.

sA ∼= C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xyC[x]⊕ xzC[x]⊕

xzC[z]⊕ yzC[y]⊕ yzC[z]⊕ xyzA

tδ2(A3) ∩ C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xyC[x]⊕

xzC[x]⊕ xzC[z]⊕ yzC[y]⊕ yzC[z] ∼= 0A

Psq ♠♦r♣s♠

A×A → A, (u, v) 7→ ∂u

∂z+∂v

∂y

st srt δ3(A3) ∼= xyzA,♦rs

H3P∼= C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xyC[x]⊕xzC[x]⊕ xzC[z]⊕ yzC[y]⊕ yzC[z]

♥ ♦♥t q

①♠♣s s r♦♣s ♦♦♠♦♦s P♦ss♦♥

♦rt♠qs

é♦rè♠ tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥ ♦rt♠q

strtr P♦ss♦♥ (A = C[x, y, z], x, y = 0, x, z = 0, y, z = xyz)

stH3P∼= C[y]⊕ zC[z]⊕ xC[x]⊕ xyC[y]⊕ xyC[x]⊕xzC[x]⊕ xzC[z]⊕ yzC[y]⊕ yzC[z]

tr♦sè♠ r♦♣ ♦♦♠♦♦ P♦ss♦♥ strtr P♦ss♦♥

(A = C[x, y, z], x, y = 0, x, z = 0, y, z = xyz) st

H3PS

∼= C[y]⊕ zC[z]⊕ xC[x]

♥ ♥ H3PS 6= H3

P .

♣tr

Préq♥tt♦♥ s strtrs

P♦ss♦♥

♦rt♠qs

♦♠♠r Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

qs ♣r♦♣rétés s strtrs ♦s②♠♣tqs

♦♥♥①♦♥ ♦rt♠q

♥trté s ♦r♠s ♦rt♠qs r♠és

Préq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt♠qs

qs r♠rqs sr ♦♦♠♦♦ s rétés P♦ss♦♥

♦rt♠qs

ss r♥P♦ss♦♥ ♦rt♠q

①♠♣s ♣♣t♦♥s

Préq♥tt♦♥ (C2, π = z1∂z1 ∧ ∂z2)

Préqtt♦♥ CP1 ♠♥ strtr

♥tr♦t♦♥

♥s ♣tr ♥♦s ét♦♥s s ♦♥t♦♥s ♥térté s strtrs

♦s②♠♣tqs t ♣réq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt♠qs

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

♥s tt ♣rt A és♥ ♥ èr ♦♠♠tt ♥tr sr ♥ ♦r♣s k

rtérstq 0 t I ♥ é ♣r♦♣r A ♥♥ré ♣r S = u1, ..., up ⊂ A.

qs ♣r♦♣rétés s strtrs ♦s②♠♣tqs

♦t µ ♥ strtr èr ♥rt♦s②♠♣tq sr DerA(log I).P♦r t♦t a ∈ A ①st ♥ ♥q ért♦♥ ♣r♥♣ δa t q

i(δa)µ = da.

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

P♦s♦♥s

a, b = −µ(δa, δb)

♣♦r t♦t a, b ∈ A. ♥ ♦t♥t ♥s ♥ r♦t P♦ss♦♥ −,− sr A. ♣s

♣♦r t♦t ui ∈ S, ①st ♥ ♥q δui ∈ Derk(log I) t q

iδuiµ =

duiui.

Psq dui ∈ ΩA ⊂ ΩA(log I), ①st δui t q iδuiµ = dui.

♥ ♦♥sèr r♦t s♥t

a, bsing :=

1

uvu, v a = u, b = v ∈ S

1

uu, b a = u ∈ S, b ∈ A− S

a, b a, b ∈ A− S

Pr♦♣♦st♦♥ ♦t strtr èr ♥rt ♦s②♠♣tq µ sr

A, ♥t ♥s A ① strtrs èrs −,− t −,−sing é♥s

♦♠♠ sss s strtrs ér♥t s ♣r♦♣rétés s♥ts

i(δu,v−uvδu,vsing)µ = u, v

(du

u+dv

v

),

uv, asing = u+ v, asing; ∀a ∈ A− I, a, b = δa(b),

[δa, δb] = δa,b,

δu,v = uv[δu, δv] + u, v(δv + δu).

Pr P♦r t♦s u, v ∈ I, ♦♥

i(δu,v−uvδu,v)µ = iδu,vµ− ıuvδu,vµ

= du, v − uvd(1

uvu, v)

= u, v(du

u+dv

v

).

♦ù ♣r♦♣rété

P♦r q st ♣r♦♣rété ♥♦s r♠rq♦♥s q

i(uv[δu,δv ]+u,v(δu+δv))µ = uvi([δu,δv ])µ+ u, v(du

u+dv

v

)µ

= uvi[δu,δv ]µ+ i(δu,v−uvδu,vsing

)µ

= i(uv[δu,δv ]+δu,v−uvδu,vsing

)µ.

st ♠♦♥trr q

i(uv[δu,δv ])µ = i(uvδu,vsing

)µ.

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

ri(uv[δu,δv ])µ = uvi([δu,δv ])µ

= uv[Lδu , iδv ]µ= uv

(Lδuiδvµ− iδvLδuµ

)

= uvd

(1

uvu, v

).

Pr rsi(uvδu,vsing

)µ = uvi(δu,vsing

)µ

= uvd (u, vsing)= uvd

(1

uvu, v

).

♦ù été ré t ♣r♦♣rété st ♥s é♠♦♥tré

♦♥♥①♦♥ ♦rt♠q

♦♥t L,L′ t L′′ tr♦s èrs ♥rt

L′′ st ♥ ①t♥s♦♥ L ♦♥ L′ s ①st ♥ st ①t ♦rt

0 // L′f // L

g // L′′ // 0

èrs ♥rt

♦t ①t♥s♦♥ t②♣ ♥t ♥ ♣♣t♦♥ ♥ér ω : L′′ → L t q

g ω = id

①t♥s♦♥ st s♥é s ω st ♥ ♦♠♦♠♦r♣s♠ èrs ♥rt

♦t ①t♥s♦♥ t②♣ ♥t ① ♣♣t♦♥s

α : L′′ −→ EndK(L′)

x 7→ αx : y 7→ [ω(x), y]

Ω :2∧L′′ −→ L′

(x; y) 7→ [ω(x), ω(y)]− ω([x, y])

ts q

[αx;αy]− α[x,y] = [ω(x, y),−]

∑

cyclerx,y,z

(αxω(x, y)− ω([x, y], z)) = 0

é♦rè♠ ❬s♠♥♥ ❪❬é♦rè♠ ❪

♦♥t L′ t L′′ ① èrs ♥rt L′ é♥♥ t s♦t : L′′ →End(L′) ♥ strtr ♠♦ ♥rt L′′ sr L′. ♦rrs♣♦♥♥

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

q à t♦t ss s♦♠♦r♣s♠ ①t♥s♦♥ t②♣ ss♦ ss Ω ∈Lalt2A(L′′, L′), st ♥ t♦♥ ♥tr ♥s♠ s sss ①t♥s♦♥s s♥és

L′ ♣r L′′ t H2(LaltA(L′′, L′)).

♦♥t L ♥ èr ♥rt t M ♥ A♠♦

❯♥ L♦♥♥①♦♥ ♥sM st ♥ ♣♣t♦♥ k♥ér ∇ : L→ End(M) t q

∇(aα)(m) = a(∇(α))(m)

∇(α)(am) = a∇(α)(m) + (ρL(α))(a)m.

s DerA(log I)♦♥♥①♦♥s sr M s♦♥t ♣♣és ♦♥♥①♦♥s ♦rt♠qs ♦♥

I sr M.

♦t L♦♥♥①♦♥ ∇ sr M ♥t ♥ ♣♣t♦♥ A♥ér

∇ :M → HomA(L,M) é♥ ♣r

∇α(m) := (∇(α))(m).

♣s ∇ ♥t sr LaltA(L,M) ♦♣értr

(d∇f)(α0, ..., αp) =i=p∑i=0

(−1)i∇αif(α0, ..., αi, ..., αp)+∑i<j

(−1)i+jf([αi, αj ], α0, ..., αi, ..., αj , ..., αp).

s♥st q ♣♦r t♦t L♦♥♥①♦♥ ∇ sr M,

(d∇f)(α0, α1) = ∇α0(f(α1))− ∇α1(f(α0))− f([α0, α1]

= (∇(α0))(f(α1))− (∇(α1)(f(α0)))− f([α0, α1]);

♣♦r t♦t α1, α2 ∈ L.

♥ ♥ ét q

(d∇∇(m))(α0, α1) = ∇α0(f(α1))− ∇α1(f(α0))− f([α0, α1]

d∇ d∇(m)(α0, α1) = (∇(α0))(f(α1))− (∇(α1)(f(α0)))− f([α0, α1])

= (∇(α0))(∇(m)(α1))− (∇(α1)(∇(m)(α0)))− ∇(m)([α0, α1])

= (∇(α0))(∇(α1)(m))− (∇(α1)(∇(α0)(m)))−∇([α0, α1])(m)

= ((∇(α0))(∇(α1))− (∇(α1)(∇(α0)))−∇([α0, α1]))(m)

= ([∇(α0),∇(α1)]−∇([α0, α1]))(m).

♥ ♥ ét ♣♣t♦♥ ♥ér ♥ts②♠étrq

ΩM L× L → End(M)

(α1, α2) 7→ [∇(α0),∇(α1)]−∇([α0, α1]

é♥t♦♥ ΩM st ♣♣é ♦rr L♦♥♥①♦♥ ∇ sr M.

♦t♦♥s Pic(A) r♦♣ s sss s♦♠♦r♣s♠s A♠♦s ♣r♦ts

r♥ 1.

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

é♦rè♠ ❬s♠♥♥ ❪ P♦r t♦t èr ♥rt L, ♣

♣t♦♥

C : Pic(A) → H2(LaltA(L,A))

M 7→ [ΩM ]

st ♥ ♦♠♦♠♦r♣s♠ A♠♦s

P♦r L = DerA(log I) té♦rè♠ ♠♣q q ♣♣t♦♥

C : Pic(A) → H2(LaltA(DerA(log I),A))

M 7→ [ΩM ].

st ♥ ♠♦r♣s♠ A♠♦s ♥s s ΩM st ♥ ♦r♠ ♦rt♠q

♦♥ I♦♥t X ♥ rété ♦♠♣① ♠♥s♦♥ n s r♠s ♦♥t♦♥s

♦♦♠♦r♣s OX t D ♥ sr rét t r X.

♦s ♥t♦♥s t♦t ré ♥ r♦t ♦♠♣① p : L→ X s F := F(L)

OX ♠♦ ss st♦♥s

♦t F ♥ ♥ ré ♥ r♦t ♦♠♣① sr X ❯♥ ♦♥♥①♦♥ ♥s F à ♣ôs ♦

rt♠qs ♦♥ D st ♥ ♦♠♦♠♦r♣s♠ C−♥ér

: F → Ω1X(logD)⊗F

ér♥t rè ♥③ s♥t

∇(fs) = df ⊗ s+ f∇(s)

Pr♦♣♦st♦♥ ♦t ♦♥♥①♦♥ ♥s F , ♦rt♠q ♦♥ D st ♥

DerX(logD)♦♥♥①♦♥ ♥s F .

♥s st ♦rr t♦t ♦♥♥①♦♥ ♦rt♠q ∇ sr F sr ♥♦té K∇.

♦t ∇ ♥ ♦♥♥①♦♥ ♦rt♠q sr F (Ui)1≤i≤n ♥ r♦r♠♥t à s ♦

rts X. ♦t s0 ∈ H0(Ui,F) t q 0 /∈ s0(Ui). ①st σ ∈ H0(Ui,Ω1X(logD))

t q ∇s0 = σ ⊗ s0, ♦rs K∇ = dσ.

♠♠ ♦t F ♥ ré ♥ r♦t ♦♠♣① sr X t ♥ ♦♥♥①♦♥ ♦

rt♠q sr F ♦rs ♣♦r t♦t ♦r♠ r♠é τ ∈ H0(X,Ω1

X(logD)), +τ⊗id

st ♥ ♦♥♥①♦♥ ♦rt♠q sr F ♦rr K = K.

Pr ♣♣♦s♦♥s q ∇ st é♥ ♣r (s) = σ⊗ s ♣♦r t♦t st♦♥ ♥♦♥ ♥

s F

♦rs(∇+ τ ⊗ id)(s) = ∇(s) + τ ⊗ s

= σ ⊗ s+ τ ⊗ s

= (σ + τ)⊗ s

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

t(∇+ τ ⊗ id)(fs) = ∇(fs) + τ ⊗ id(fs)

= df ⊗ s+ fσ ⊗ s+ fτ ⊗ s

= df ⊗ s+ f(∇+ τ ⊗ id)s.

D st ♥ sr à r♦s♠♥ts ♥♦r♠① ♦rs ①st ♥ s②stè♠ ♦♦r♦♥♥és

(zi)1≤i≤n X ♥ t♦t ♣♦♥t p D t q

σ =r∑

i=1

aidzi

zi+

n∑

i=r+1

aidzi

♦ù ai ∈ H0(X,OX).

♠♠ ♦♥t D ♥ sr à r♦s♠♥ts ♥♦r♠① t α ∈H0(X,Ω1

X(logD)) dα = 0 ♦rs rés α st ♦♥st♥t sr t♦t

♦♠♣♦s♥t ♣rt s♥èr D. ♥ s ♦r♠s ②♥t ♠♦♥s ♥

rés ♥♦♥ ♥ ♠t r♣rés♥tt♦♥ s♥t

α =r∑

j=1

αidfjfj, α1, ..., αr ∈ C.

♠♠ ♥♦s é♠♦♥tr♦♥s ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ♦t D ♥ sr à r♦s♠♥ts ♥♦r♠① X t ♥ ♦♥

♥①♦♥ à ♣ôs ♦rt♠qs ♦♥ D F . ♦rr K∇ st ♥ s

t s♠♥t s ♦r♠ ♦♥♥①♦♥ ss♦é st ♦r♠ σ =r∑i=1aidzi

zi

ai ∈ C.

rt♦♥ ♠♦♥tr q t♦t ♦♥♥①♦♥ ∇ sr F , ♦rt♠q ♦♥ D

ér rt♦♥ s♥t

∇δ(fs)− f∇δs = δ(f)s

♣♦r t♦s s ∈ M, f ∈ OX t δ ∈ DerX(logD).

♥ s♣♣♦s q h st ♦♥t♦♥ é♥t♦♥ D t ♦♥ r♣♣ q♥ ♦♣ér

tr ér♥t ϕ ♦rr r sr F st t ♦rt♠q ♦♥ D s s 7→[ϕ(hs) − hϕ(s)]h−1 st ♥ ♦♣értr ér♥t ♦rr (r − 1) sr F . ♥ ♥♦t

+1 (logD) ♠♦ s ♦♣értrs ér♥ts ♦rr ≤ 1, ♦rt♠qs

♦♥ D sr F .P♦r t♦t ♦♥♥①♦♥ ♦rt♠q ∇ t t♦t δ ∈ DerX(logD) ♦♥ ∇δ ∈+

1 (logD).

♠♠ ♦t ϕ ♥ ♦♣értr ér♥t ♦rt♠q ♣r♠r ♦rr P♦r

t♦t f ∈ OX , ①st ♥ ♥q f ∈ OX t q [ϕ(fs)− fϕ(s)] = f s.

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

Pr P♦r t♦t ♦♣értr ér♥t ♦rt♠q ♣r♠r ♦rr ϕ, [s 7→ϕ(fs)− fϕ(s)] ∈ +

0 (logD). ①st f ∈ OX t q [ϕ(fs)− fϕ(s)] = f s. g

st ♥ tr éé♠♥t OX t q [ϕ(fs) − fϕ(s)] = gs, ♦rs f s = gs ♣♦r t♦t

s ∈ E ♦♥ f = g.

♦r♦r ϕ st ♥ ♦♣értr ér♥t ♦rt♠q ♣r♠r ♦rr

♦♥ D ♦rs h ∈ hOX

Pr P♦r t♦s s ∈ F , ϕ(hs)− hϕ(s) = hs t ①st g ∈ OX t q ϕ(hs)−hϕ(s) = hgs Pr st (h− hg)s = 0 ♣♦r t♦t s.

s♥st q t♦t ♦♣értr ér♥t ϕ ♦rt♠q ♣r♠r ♦rr ♦♥

D, ♦♥♥ à ♥ ♣♣t♦♥ σϕ : OX → OX é♥ ♣r σϕ(f) = f t q

[ϕ(fs)− fϕ(s)] = f s ♣♦r t♦t s ∈ F .

♠♠ P♦r t♦t ϕ ∈ +1 (logD), σϕ ∈ H0(X,Der1X(logD))

Pr ♦♥t f, g ∈ OX t s ∈ F . ♥

σϕ(f.g)s = ϕ(f(gs)− fgϕ(s)

= σϕ(f)(gs) + fϕ(gs)− fgϕ(s)

= σϕ(f)(gs) + f(ϕ(gs)− gϕ(s))

= (σϕ(f)g + fσϕ(g))s

tr ♣rtσϕ(h)s = ϕ(hs)− hϕ(s)

= hmh(s)

♦♥ (σϕ(h)− hmh)s = 0 ♣♦r t♦t s.

Pr st σϕ(h) ∈ hOX σϕ ∈ H0(X,Der1X(logD)).

Pr♦♣♦st♦♥ +1 (logD) st st ♣♦r ♦♠♠ttr

Pr ♦t ϕ1, ϕ2 ① éé♠♥ts +1 (logD) ♥

ϕ1ϕ2(fs) = ϕ1

(fϕ2(s) + f2s

)

= fϕ1(fϕ2(s) + ϕ1(f2s))

= fϕ1 (ϕ2(s)) + f1ϕ2(s) + f2ϕ1(s) +¯f2

1.s

ç♦♥ ♥♦

ϕ2ϕ1(fs) = fϕ2 (ϕ1(s)) + f2ϕ1(s) + f1ϕ2(s) +¯f1

2s

♣r st

ϕ1ϕ2(fs)− ϕ2ϕ1(fs)− f (ϕ1ϕ2 − ϕ2ϕ1) (s) = ( ¯f21 − ¯f1

2)s.

Pr rs ♣♦r t♦t ϕ1, ϕ2 ∈ +1 (logD), ①st h1, h2 ∈ OX ts q

[ϕ2(hs)− hϕ2(s)]1h = h2s t [ϕ1(hs)− hϕ1(s)]

1h = h1s

h2 = hh2 t h1 = hh1.

♠ê♠ ①st h21, h12 ∈ OX ts q ¯h12= hh12 t ¯h2

1= hh21.

Pr st

ϕ1ϕ2(hs)− ϕ2ϕ1(hs)− h (ϕ1ϕ2 − ϕ2ϕ1) (s) = (¯h21 − ¯h1

2)s = h[h21 − h12]s

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

♥trté s ♦r♠s ♦rt♠qs r♠és

♦t X ♥ rété ♦♠♣① ♠♥s♦♥ 2n D ♥ sr rét X.

♠♠ D stst ♣r♦♣rété érè♠ ♦rs ♣♦r t♦t

ω ∈ Ω2X(logD), ♦♥ res(ω) ∈ ΩX .

Pr é♦ éq♥ s ♣r♦♣rétés t té♦rè♠

♥s st ♥♦s s♣♣♦s♦♥s q D stst ♣r♦♣rété té♦rè♠

és♥♦♥s ♣r HkDR−Log(X) kime r♦♣ ♦♦♠♦♦ ♠ ♦rt

♠q X. ♦s ♦♥s st ♠♦r♣s♠s r♦♣s s♥t

... // H∗(X,Z)i // H∗(X,C)

∼=

p // H∗(X,Ω∗X(logD)) // ...

H∗(X,Ω∗X)

♦t [ω] ♥ éé♠♥t H2(X,Ω∗X(logD))

é♥t♦♥ ω st t ♥tér s [ω] ♣♣rt♥t à ♠ p i. ♣r♦♣♦st♦♥ s♥t ♥♦s ♦♥♥ ♥ rtérst♦♥ s ♦r♠s ♦rt♠qs

r♠és

Pr♦♣♦st♦♥ ♦t ω ♥ st♦♥ Ω2X(logD). ♥

d(ω) = 0 s t s♠♥t s s ♦r♠ rés t s ♣rt ss s♦♥t t♦ts r♠és

Pr ♥t t q D ér s ♣r♦♣rétés é♦rè♠ t

♦♥ ω =dh

h∧ res(ω) + ωl ♦ù res(ω) st ♦r♠ rés ss♦é à ω t ωl st

♣rt ss ω.

é♦rè♠ ♦t ω ♥ ♦r♠ r♠é ♦rt♠q ♦♥

s ♣r♦♣rétés s♥ts s♦♥t éq♥ts

ω =dh

h∧ ψ + η st ♥tér

res(ω) st ①t t ①st [ω0] ∈ H2(X,C) ♥tér t q

[ω0] = [η].

Pr ω st ♥tér ♦rs ①st [ω1] ∈ H2(X,Z) t q [ω] = p i[ω1].

♦♥tr♦♥s q [ω0] = i([ω1]).

Psq ω st ♥tér ①st [ω1] ∈ H2(X,Z) t q [ω] = p i[ω1]. tr♠♥t

t ①st ♥ ♦r♠ ♦rt♠q α = α0dh

h+ α1 t q ω− ω0 = dα. ♦♥

−dα0 = ψ t η = ω0 + dα1.

é♣r♦q♠♥t s ω0 + dλ = η t ψ = dβ ω0 ♥tr ♦rs

ω = d(−β dhh) + η

= ω0 + dλ+ d(−β dhh)

= ω0 + d(λ− βdh

h)

Préq♥tt♦♥ s strtrs ♦s②♠♣tqs

Pr st [ω] = [η] = [ω0].

s tr① ♦st♥t ♥s ❬♦st♥t ❪ t ♦r ♥s ❬♦r ❪ r

♣♦s♥t sr ♣r♥♣ q♥tt♦♥ ♣r♦♣♦sé ♣r r ♥s ❬r ❪

♣r♥♣ ♣r♠t ♠♦ésr ♠té♠tq♠♥t q s ♣②s♥s ♣♣♥t

q♥tt♦♥ st sé sr ♦♥strt♦♥ ♥ s♦♠♦r♣s♠ ♥tr èr

s ♦♣értrs sr ♥ s♣ rt H t èr s ♦srs s

sqs F(X) ♦♥sttés s ♦♥t♦♥s é♥s ♥s ♥ rété s②♠♣tq (X,ω).

Ps ♣résé♠♥t s ϕ st ♥ t s♦♠♦r♣s♠ rt stsr s ♣r♦♣rétés

s♥ts

ϕ st t

s f st ♥ ♦sr ♦♥st♥t ♦rs ϕ(f) st ♠t♣t♦♥ ♣r f.

[f1, f2] = f3 ♦rs ϕ(f1)ϕ(f2) − ϕ(f2)ϕ(f1) = −ihϕ(f3) ♦ù h és♥ ♦♥

st♥t P♥

q éqt à ①st♥ ♥ r♣rés♥tt♦♥ ϕ (F(X), ω) r♥♥t ♦♠♠

tt r♠♠ èrs ♥rt s♥t

0 // F(X)m // +

1 (Γ(L))σ // DerX // 0

0 // R //

OO

(F(X), ω)

ϕ

OO

// Ham(F(X))

OO

// 0

ϕ é♥ ♣r

ϕ(as) = ∇v(a)s+ 2iπas

♦r ❬❯r♥ ❪ ♦ù ∇ st ♥ ♦♥♥①♦♥ ♥s ♥ ré ♥ r♦t ♦♠♣① L sr

X t Ham(F(X)) st èr s ♠♣s ♦♠♥t ♠t♦♥♥s

♦rsq ♦♥ r♠♣ rété s②♠♣tq (X,ω) ♣r ♥ rété ♦s②♠♣tq

(X,ω,D), ①è♠ ♥ r♠♠ st r♠♣é ♣r

0 // C // (OX , ω) // HωX(OX) // 0

♥♦s ♠♥t♥♦♥s ①♣rss♦♥ ϕ ♦♥♥é ♣r ♦rs ♣♦r t♦s f, g ∈H0(X,OX) t s ∈ E , ♦♥

ϕ(f)ϕ(g)s = ϕ(f)(ϕ(g)s)

= ϕ(f)[∇v(g)s+ 2πigs]

= ∇v(f)(∇v(g)s+ 2πigs) + 2πi(f∇v(g)s+ 2πifgs)

= ∇v(f)∇v(g)s+ 2πi∇v(f)(gs) + 2πi∇v(g)s− 4π2fgs

= ∇v(f)∇v(g)s+ 2πi(H(df).g)s+ 2πig∇v(f)s+ 2πif∇v(g)s− 4π2fg

♥ é♥♥t s rôs f t g ♦♥ ♦t♥t

ϕ(g)ϕ(f)s = ∇v(g)∇v(f)s+ 2πi(H(dg).f)s+ 2πig∇v(g)s+ 2πig∇v(f)s− 4π2gfs

♣r st

[ϕ(f), ϕ(g)]s = [∇v(f),∇v(g)]s+ 4πiω(v(f), v(g))s

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

tr ♣rt

ϕ(f, g) = ∇v(f,g)s+ 2πif, gs= ∇[v(f),v(g)]s+ 2πif, gs= [∇v(f),∇v(g)]−K∇(v(f), v(g))s+ 2πif, gs= [ϕ(f), ϕ(g)]s+ 2πif, gs−K∇(v(f), v(g))s

♥s s ♣r♦♣rété ♣r♥♣ r st stst s t s♠♥t s

K∇ = 2πiω

♥ ♥s ♣r♦é ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ ❯♥ rété ♦s②♠♣tq (X,ω,D) st ♣réq♥t s t

s♠♥t s ①st ♥ ré ♥ r♦t ♦♠♣① sr X ♣♦ssé♥t ♥ ♦♥♥①♦♥

♦rt♠q ♦♥ D ♦rr 2iπω.

Préq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt

♠qs

♥s tt ♣rt (X,D,Υ) és♥r ♥ rété P♦ss♦♥ ♦rt♠q

♦♥ ♥ sr rét t r D X t♥sr P♦ss♦♥ ss♦é Υ.

qs r♠rqs sr ♦♦♠♦♦ s rétés P♦ss♦♥ ♦rt♠qs

♥ ♥♦t ∂D ér♥t P♦ss♦♥ ♦rt♠q Υ. ss ♦♦

♠♦♦ P♦ss♦♥ ♦rt♠q ♥ ♦② P sr ♥♦té [P ]D.

é♥t♦♥ ∂D t ér♥t d ♠ ♦rt♠q ♦♥

♠♠ s♥t

♠♠ ♣♣t♦♥ H ér

∂D H = −H d

♥ ♥ ét ♣r♦♣♦st♦♥ s♥t

Pr♦♣♦st♦♥ H∗DR−Log(X) st r♦♣ ♦♦♠♦♦ ♠ ♦

rt♠q X ♦rs H : (Ω∗X(logD), d) → (Der∗X(logD), ∂log) ♥t ♥ ♠♦r

♣s♠ é♥ ♣r

H : H∗DR−Log(X) → H∗

PS(X)

[α] 7→ [H(α)]D

Préq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt♠qs

ss r♥P♦ss♦♥ ♦rt♠q

♦t (X,D,Υ) ♥ rété P♦ss♦♥ ♦rt♠q p : L → X ♥ ré ♥

r♦t ♦♠♣① sr X t Γ(L) s♦♥ ♠♦ st♦♥s

é♥t♦♥ ❯♥ ért♦♥ ♦rt♠q ♦♥trr♥t Dlog sr p : L → X

st ♥ ♣♣t♦♥ Dlog : C♥ér Ω1X(logD) → EndC(Γ(L)) t q

Dlogα (fs) = fDlog

α s+ (H(α)f)s

♣♦r t♦t α ∈ Ω1X(logD) t s ♥ st♦♥ ♦ Γ(L).

Dlog st t ♦♠♣t ♥ ♠étrq r♠t♥♥ h sr p : L→ X s ♣♦r t♦t

α ∈ ΩX(logD), s1, s2 ∈ Γ(L)

H(α)(h(s1, s2)) = h(Dlogα s1, s2) + h(s1, D

logα s2).

♠rq ∇ st ♥ ♦♥♥①♦♥ ♦rt♠q sr p : L → X ♦rs

Dα = ∇H(α) st ♥ ért♦♥ ♦rt♠q ♦♥trr♥t sr p : L→ X

é♥t♦♥ ♥ ♣♣ ♦rr ♥ ért♦♥ ♦rt♠q ♦♥trr♥t

Dlog sr p : L→ X t♦t ♣♣t♦♥

CD : Ω1X(logD)× Ω1

X(logD) → EndC(Γ(L))

é♥ ♣r

CD(α, β) = Dlogα Dlog

β −Dlogβ Dlog

α −Dlogα,β

♣♦r t♦s α, β ∈ Ω1X(logD).

♥ ♣r♦♣rété s♥t s ért♦♥s ♦rt♠qs ♦♥trr♥ts

Pr♦♣♦st♦♥ CD st OX♥ér ♥ts②♠étrq

Pr P♦r t♦s α, β ∈ Ω1X(logD) ♦♥

CD(β, α)s = (Dlogβ Dlog

α −Dlogα Dlog

β −Dlogβ,α)s

= −(Dlogα Dlog

β −Dlogβ Dlog

α −Dlogα,β)s

= −CD(α, β).

♦t f ♥ st♦♥ OX . ♥

CD(fα, β)s

= (Dlogfα Dlog

β −Dlogβ Dlog

fα −Dlogfα,β)s

= fDlogα Dlog

β s−Dlogβ (fDαs)−Dlog

fα,β+(H(β)f)αs

= fDlogα Dlog

β s− fDlogβ (Dαs)− (H(β)f)Dlog

α s− fDlogα,βs+ (H(β)f)Dlog

α s

= f(Dlogfα Dlog

β −Dlogβ Dlog

fα −Dlogfα,β)s

= fCD(α, β)s.

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

Pr♦♣♦st♦♥ ♦t p : L→ X ♥ ré ♥ r♦t ♦♠♣① sr (X,D,Υ) ♠♥

♥ ért♦♥ ♦rt♠q ♦♥trr♥t Dlog ♦rr CD. ♦rs

CD é♥t ♥ ss ♦♦♠♦♦ [CD]D ♥s H2

PS(X),

[CD]D ♥ é♣♥ ♣s Dlog,

♣s Dlog st ♦♠♣t ♠étrq r♠t♥♥ h sr p : L →X ♦rs CD = −CD.

Pr ♦t s ♥ st♦♥ p : L → X ♥ s♥♥♥t ♣s sr X. Psq

q r p : L → X st ♥♠♥s♦♥♥ ♦rs té ♥tr Ω1X(logD)

t Der1X(logD)) ♠♣q q ♣♣t♦♥ α 7→ Dαss st C♥ér ①st ♦♥

♥ ♥q ♠♣ trs ♦rt♠q δ sr X t q

Dlogα s = 〈α, δ〉s

♦ù 〈−,−〉 és♥ r♦t té ♥tr (Ω1X(logD) t Der1X(logD)).

♣s ♣♦r t♦s α, β ∈ ΩX(logD) ♦♥

CDlog(α, β)s = (Dlogα Dlog

β −Dlogβ Dlog

α −Dlogα,β)s

= Dlogα (〈β, δ〉s)−Dlog

β (〈α, δ〉s)− 〈α, β, δ〉s= 〈α, δ〉〈β, δ〉s+ H(〈α, δ〉)s− 〈β, δ〉〈α, δ〉s− H(〈β, δ〉)s− 〈α, β, δ〉s= H(α)(〈β, δ〉)s− H(β)(〈α, δ〉)s− 〈α, β, δ〉s= ∂Dδ(α, β)s

♦♥ CDlog = ∂Dδ. t ♣r st ∂DCDlog = ∂2Dδ = 0 ♦♥ CD st ♥ ♦②

P♦ss♦♥ ♦rt♠q

♦t D′ ♥ tr ért♦♥ ♦rt♠q ♦♥trr♥t sr p : L → X

♦rr ss♦é C ′D t δ′ ♠♣ trs ♦rt♠q ss♦é

♥ CD′ − CDlog = ∂Dδ′ − ∂Dδ CD′ = CDlog + ∂D(δ

′ − δ).

Pr rs ♣♦r t♦t α ∈ Ω1X(logD) ♦♥ D′

α −Dα ∈ EndC(Γ(L))

①st ♦♥ ♥ ♠♣ trs ♦rt♠q δ′′ t q ♣♦r t♦t s ∈ Γ(L)

(D′α −Dlog

α )s = 〈α, δ′′〉s♦♥ 〈α, δ”〉s = D′

αs−Dlogα s = 〈α, δ′〉−〈α, δ〉s 〈α, δ”〉 = 〈α, δ′−δ〉 δ′′ = δ′−δ

t ♦♥ CD′ = CDlog +∂D(δ′− δ) = CDlog +∂Dδ” stàr [CD′ ]log = [CDlog ]log.

♣♣♦s♦♥s q Dlog st ♦♠♣t ♥ ♠étrq r♠t♥♥ h sr p :

L→ X t s♦t (e) ♥ s ♦rt♦♦♥ ♦ Γ(L) ♦rs α ∈ Ω1X(logD).

♥ ♦♥

H(α)(h(e, e)) = h(Dlogα e, e) + h(e,Dlog

α e) 0 = h(〈α, δ〉e, e) + h(e, 〈α, δ〉e)

〈α, δ〉 + 〈α, δ〉 = 0 δ + δ = 0. s♥st q δ t ♦♥ CDlog = ∂Dδ s♦♥t

♠♥rs ♣rs

♣rt ♣r♦♣rété iii) té♦rè♠ s♥st q 12πi [CDlog ]D ∈ H2

PS(X)

♥ ♥ ét é♥t♦♥ s♥t

é♥t♦♥ 12πi [CDlog ]D st ♣r♠èr ss r♥P♦ss♦♥ ♦rt♠q

p : L→ X.

Préq♥tt♦♥ s strtrs P♦ss♦♥ ♦rt♠qs

♦s ♦♥s à ♣rés♥t étr ♥ ♥ ♥tr ♣r♠èr ss r♥ C1(L) ♥

ré ♥ r♦t ♦♠♣① r♠t♥ p : L → X sr ♥ rété P♦ss♦♥ ♦

rt♠q (X,D,Υ) t s ss r♥P♦ss♦♥ ♦rt♠q 12πi [CDlog ]D. ♦s

s♣♣♦s♦♥s q♥ ♣s D stst s ♣r♦♣rétés s♥ts

D st à r♦s♠♥t ♥♦r♠①

D = ∪j∈IDj é♦♠♣♦st♦♥ ♥ ♦♠♣♦s♥ts rréts D, ♦rs

q Dj st ss I és♥ ♥s♠ s ♥s

♦t ∇ ♥ ♦♥♥①♦♥ ♦rt♠q sr ♥ ré ♥ r♦t ♦♠♣① r♠t♥ L sr

X. ♦r♠ ♦♥♥①♦♥ ♦rt♠q α0 é♥ ♣r rt♦♥ ∇δs = 〈α0, δ〉sér dα0 = K.

♥ ♦♥

c1(L) = [i

2πK]D = [

i

2πdα0].

♥ ♣♦s ♣♦r t♦t α ∈ Ω1X(logD) Dα := ∇H(α).

♦t δ ♥ ♠♣ trs ♦rt♠q é♥ ♣r rt♦♥ ♥

Dαs = ∇H(α)s⇔ 〈α, δ〉s = 〈α0, H(α)〉s⇔ 〈α, δ〉s = −〈α, H(α0)〉 ⇔ δ = −H(α0).

Pr st CDlog = ∂Dδ = −∂DH(α0) = H(dα0).

♥ ♦♥ [i

2πCDlog ]D = [

i

2πH(dα0)] = H([

i

2π]dα0) = H(c1(L)).

tr♠♥t t s sss r♥P♦ss♦♥ ♦rt♠q t r♥ ré

♥ r♦t ♦♠♣① r♠t♥ L s♦♥t és ♣r rt♦♥

[i

2πCDlog ]D = H(c1(L)).

♦t p : L → X ♥ ré ♥ r♦t ♦♠♣① r♠t♥ ♠♥ ♥ ért♦♥ ♦♥

trr♥t ♦rt♠q Dlog ♦♥ ♥ sr D stss♥t s ②♣♦tèss

t sss ♣rès ♣r♥♣ r ♥s ❬r ❪ ♠♦r

♣s♠

ϕ : OX → EndC(Γ(L))

é♥ ♣r

ϕ(f)s = Dlogdf s+ 2πifs

♦t êtr ♥ r♣rés♥tt♦♥ èr (OX , −,−Υ) ♣r Γ(L). q ♠♣q q

CDlog = −2πiΥ

♥ ♥ ét q

Pr♦♣♦st♦♥ ♦♣értr ϕ st ♥ ♦♠♦♠♦r♣s♠ èrs s t

s♠♥t s CDlog = −2πiΥ

♦s ♦♣t♦♥s é♥t♦♥ s♥t

é♥t♦♥ ❯♥ rété P♦ss♦♥ ♦rt♠q (X,D,Υ) st t ♦ ♣réq♥

t s ①st ♥ ré ♥ r♦t ♦♠♣① r♠t♥ p : L → X ♣♦r q

♦♣értr ϕ é♥ ♣r st ♥ é♥ t st ♥ ♠♦r♣s♠ èrs

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

Pr♦♣♦st♦♥ ♦t (X,D,Υ) ♥ rété P♦ss♦♥ ♦rt♠q ♦♥

♥ sr D stss♥t s ②♣♦tèss t (X,D,Υ) st ♦ ♣réq♥

t s ①st ♥ ♠♣ trs ♦rt♠q δ t ♥ ♦r♠ ♦rt♠q

ω ♥tér t q

Υ+ ∂Dδ = H(ω).

Pr ♣♣♦s♦♥s (X,D,Υ) ♦ ♣réq♥t t ♥♦t♦♥s CDlog ♦rr

ért♦♥ ♦♥trr♥t Dlog ss♦é ré ♣réq♥tq L→ X ♦rrs♣♦♥♥t

♦rsi

2πCDlog = Υ ♦t K∇ ♦rr ♦♥♥①♦♥ r♠t♥♥ ∇ sr L.

♣rès ♦♥ c1(L) = [i

2πK∇]. ♥ ♣r♥ ω =

i

2πK∇ Pr rs ∇ ♥t

♥ ért♦♥ ♦rt♠q ♦♥trr♥t D é♥ ♣r Dα = ∇H(α) ♣♦r t♦t

α ∈ Ω1X(logD). ♥ ♥♦t CD s ♦rr ♣rès rt♦♥ ♦♥

H([ω]) = [i

2πCD]

r à éqt à [H(ω)] = [i

2πCD]

D. ♥ rt Pr♦♣♦st♦♥ ♦♥ ♥

ét q [CDlog ]D = [CD]D. q ♠♣q q ①st ♥ ♠♣ trs

♦rt♠q λ t qi

2πCDlog =

i

2π∂Dλ+

i

2πCD =

i

2π∂logλ+ H(ω). stàr

Υ+ ∂log(−i

2πλ) = H(ω). st ♦♥ ♣r♥r δ = − i

2πλ.

é♣r♦q♠♥t ♦♥ s♣♣♦s qs ①st♥t δ t ω ♦♠♠ ♥s s ②♣♦tèss

Pr♦♣♦st♦♥ ♦rs ♥ rt ♥térté ω ♥ ré ♥ r♦t ♦♠♣①

r♠t♥ L → X ♥ ♦♥♥①♦♥ r♠t♥♥ ♦rt♠q ∇ t ♦rr

−2πω. P♦s♦♥s Dlog(α)s = ∇H(α)s+ 2πi〈δ, α〉 t ♠♦♥tr♦♥s q st ♥ ért♦♥

♦♥trr♥t ♦rt♠q ♦♥t ♦rr CDlog Dlog ér rt♦♥

st r q st ♥ ért♦♥ ♦♥trr♥t ♦rt♠q

♦♥t α, β ∈ Ω1X(logD) t s ♥ st♦♥ L. ♥

CDlog(α, β)s = (Dlogα Dlog

β −Dlogβ Dlog

α −Dlog[α,β])s

= Dlogα (∇H(β)s+ 2πi〈β, δ〉)s−Dlog

β (∇H(α)s+ 2πi〈α, δ〉s)− ∇H[α,β]s− 2πi〈[α, β], δ〉s= ∇H(α)((∇H(β)s+ 2πi〈β, δ〉)s) + 2πi〈α, δ〉((∇H(β)s+ 2πi〈β, δ〉)s)− ∇H(β)((∇H(α)s+ 2πi〈α, δ〉)s)− 2πi〈β, δ〉((∇H(α)s+ 2πi〈α, δ〉)s)− ∇[H(α),H(β)]s− 2πi〈[α, β], δ〉s=

(∇H(α)∇H(β)s−∇H(β)∇H(α)s−∇[H(α),H(β)]s

)

+ 2πi(H(α)〈β, δ〉)s− H(β)〈α, δ〉)s− 〈[α, β], δ〉s

)

= −2πi (ω) (H(α), H(β))s+ 2πi∂Dδ (α, β) s

= −2πiΥ(α, β)s

♣♦r α, β ∈ Ω1X(logD) t ♣♦r t♦t st♦♥ ♦ s L.

P♦s♦♥s h = h1...hk é♦♠♣♦st♦♥ h ♥ ♦♠♣♦s♥ts rréts Di D

①♠♣s ♣♣t♦♥s

é♥s ♣r s hi. ♣rès Pr♦♣♦st♦♥ s ♦r♠s ♦rt♠qs ♥té

rs s♦♥t à réss ①ts t à ♦r♠ ♦♦♠♦r♣ ss♦é r♠é t ♥tér

ω st ♥tér ♦rs H(ω) =k∑i=1

Ri

hi(H(dhi)) +H(ω0) R

i st♦♥ ♦ OX

t ω0 ♥ ♦r♠ ♦♦♠♦r♣ ♥tér ♥ ♥ ét ♦r♦r s♥t

♦r♦r ♦t (X,D,Υ) ♥ rété P♦ss♦♥ ♦rt♠q ♦♥ ♥

sr D stss♥t s ②♣♦tèss é♦rè♠ (X,D,Υ) st ♦

♣réq♥t s ①st ♥ ♠♣ trs ♦rt♠q δ s ♦♥t♦♥s ♦♦

♠♦r♣s Ri, i = 1, ..., k t ♥ ♦r♠ ω0 ♦♦♠♦r♣ sr ♥ s♦s rété

♠♥s♦♥ ♥ X, ♥tér t q

Υ+ ∂D(δ −k∑

i=1

Ri

hi(H(dhi))) = H(ω0)

①♠♣s ♣♣t♦♥s

Préq♥tt♦♥ (C2, π = z1∂z1 ∧ ∂z2)

P♦s♦♥s X = C2;D = (0, z), z ∈ C ♦s s♦♥s q ω0 = dz1 ∧ dz2 st ♥

strtr s②♠♣tq sr C2 ♦♥t strtr P♦ss♦♥ ss♦é st é♥ ♣r

z1, z2 = 1 ♥ ♣♦s♥t f, gnew := f, h1g, h2 − f, h2g, h1 ♦ù h1 = z1 t

h2 = z1z2 ♦♥ ♦t♥t z1, z2new = z1 q st ♥♦tr strtr P♦ss♦♥ π.♦♥tr♦♥s

q tt strtr st ♦ ♣réq♥t P♦r ♦s ♦♥s rr ♥

st♦♥ ω0 Ω2X(logD) t q

①st ♥ st♦♥ α0 ∈ Ω2X ♥tér t α0 ∈ [ω0]

H(ω0) ∈ [z1∂z1 ∧ ∂z2 ]♦♥sér♦♥s ♦♥ ♦♠♣① ♠ ♦rt♠q s♥t

K : 0 // ΩA(log I)X d0 // Ω1X(logD)2

d1 // ΩA(log I)X // 0

♦ù d0(f) := z1∂z1fdz1z1

+ ∂z2fdz2 t d1(f1dz1z1

+ f2dz2) = (z1∂z1f2 − ∂z2f1)dz1z1

∧ dz2♦rs H2(K) = 0 ♥ t ♣♦r t♦t st♦♥ g Ω1

X(logD) ①st ♥ st♦♥

f OX t q d1(fdz2) = gdz1z1

∧ dz2. s♥st q t♦t st♦♥ Ω1

X(logD) sr s♦t♦♥ ♣r♦è♠ st

♣r♥r α0 = 0.

Préqtt♦♥ CP1 ♠♥ strtr

♥s ❬♦r♦s♥ t ❪ st é♠♦♥tré q s strtrs P♦ss♦♥

②♥♥r♥ t r♦♦♥st♥t♦r s♦♥t ♦♠♣ts

sr CP1. tr♠♥t t r ♦♠♥s♦♥ ♥ér ♥t é♠♥t ♥ strtr

♣tr Préq♥tt♦♥ s strtrs P♦ss♦♥

♦rt♠qs

P♦ss♦♥ sr CP1 ♥s ❬♦r♦s♥ t ❪ t ❬♦t♦ ❪ s trs ♠♦♥

tr♥t q tt strtr st ♣r♠étré sr C ♣r

πλ := − i

2(zz + 1)(λ+ (λ+ 2)zz))

∂

∂z∧ ∂

∂z.

t q st s♥èr ♣♦r λ ∈ [−2, 0]

♦t♦ ♠♦♥tr é♠♥t ♥s ❬♦t♦ ❪ q s r♦♣s ♦♦♠♦♦ P♦s

s♦♥ tt strtr s♦♥t

H0 ∼= H1 ∼= C; H2 ∼= C2

t s λ = 0.

P♦r λ = 0 ♦♥

π0 = −i(zz + 1)z∂

∂z∧ z ∂

∂z

♦ù

z∂z :=z

2(∂x − i∂y); t z∂z :=

z

2(∂x + i∂y).

P♦r t♦ts ♦♥t♦♥s a, b ♦♥

a, b = −i(1 + zz)zz(∂a

∂z

∂b

∂z− ∂b

∂z

∂a

∂z).

st ♥ ♦rt♠q ♦♥ sr D0 := zz = 0.P♦r t♦t ♦♥t♦♥ a, ♦♥

∂0f =∂a

∂zz,− − ∂a

∂z−, z

= i(1 + zz)(z∂a

∂zz∂

∂z− z

∂a

∂zz∂

∂z).

éqt♦♥ ♦♥ ét q H0 ∼= C.

♠ê♠ ♣♦r t♦t ♠♣ trs ♦rt♠q δ = az∂

∂z+ bz

∂

∂z♦♥

∂1δ = i(1 + zz)(z∂za− z∂zb) + izz(a+ b).

Pr rs tt strtr ♥t sr X −D0 strtr s②♠♣tq é♥ ♣r

ω0 = i1

1 + zz

dz

z∧ dz

z.

♣s ♣♦r t♦ts ♦♥t♦♥s a, b ♦♥

(1 + zz)(z∂z(a)− z∂z(b)) 6= 1.

♥ t s ①st a0, b0 ts q (1 + zz)(z∂z(a0) − z∂z(b0)) = 1 ♦rs ♣♦r t♦t

z ∈ U := z ∈ C; 0 < |z| < 4 ♦♥ r za(0)− zb(0) =1

1 + zz. q st sr

r ♥tr♥rt q1

3=

1

4 ♦rsq♦♥ ♣r♥ z = 2 ♣s z = 3.

♠♦♥tr q ω0 6= 0 ∈ H2(CP1,Ω∗(logD0))

①♠♣s ♣♣t♦♥s

♥ ♣t ♦♥ ♦♥r q ω0 st ♥ strtr ♦s②♠♣tq sr CP1. Pr ♦♥

séq♥t H : Ω∗(logD0) → Der∗X(logD) st qss♦♠♦r♣s♠

t ♦♥

H2 ∼= H−1

(C[[z, z]]

〈1 + zz〉 i1

1 + zz

dz

z∧ dz

z

).

♠rq♦♥s q

H(i1

1 + zz

dz

z∧ dz

z) = −i(1 + zz)z∂z ∧ z∂z

s♥st q π0 st ♦ ♣réq♥t s t s♠♥t s

[i1

1 + zz

dz

z] = 0 ∈ H1((CP1,Ω∗(logD0)) ∼= C[[z]]

dz

z⊕ C[[z]]

dz

z.

q st sr ♦♥ π0 ♥st ♣s ♦ ♣réq♥t

♥♥①

P♦♥ts ét qqs

é♠♦♥strt♦♥s

é♠♦♥strt♦♥ ♦r♦r

♦t −,−0 ♥ strtr P♦ss♦♥ sr A = k[x, y] ♦ A = k[x1, x2, x3]

• s A = k[x, y].

♣rès ♠♠ st ♣r♦r q

a, a00b, c0 + b, a00c, a0 + c, a00a, b0 = 0

♣♦r t♦t a, b, c ∈ A.r ♣♦r t♦t f, g ∈ A,

f, g = (∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x)x, y

♦♥

a, a00 = (∂a

∂x

∂a0∂y

− ∂a

∂y

∂a0∂x

)x, y0, b, c0 = (∂b

∂x

∂c

∂y− ∂b

∂y

∂c

∂x)x, y0

b, a00 = (∂b

∂x

∂a0∂y

− ∂b

∂y

∂a0∂x

)x, y0, c, a0 = (∂c

∂x

∂a

∂y− ∂c

∂y

∂a

∂x)x, y0

c, a00 = (∂c

∂x

∂a0∂y

− ∂c

∂y

∂a0∂x

)x, y0, a, b0 = (∂a

∂x

∂b

∂y− ∂a

∂y

∂b

∂x)x, y0

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

♦♥ ♥ ét q

1

(x, y0)2a, a00b, c0 + b, a00c, a0 + c, a00a, b0 =

∂a

∂x

∂a0∂y

∂b

∂x

∂c

∂y− ∂a

∂x

∂a0∂y

∂b

∂y

∂c

∂x− ∂a

∂y

∂a0∂x

∂b

∂x

∂c

∂y+∂a

∂y

∂a0∂x

∂b

∂y

∂c

∂x+

∂b

∂x

∂a0∂y

∂c

∂x

∂a

∂y− ∂b

∂x

∂a0∂y

∂c

∂y

∂a

∂x− ∂b

∂y

∂a0∂x

∂c

∂x

∂a

∂y+∂b

∂y

∂a0∂x

∂c

∂y

∂a

∂x+

∂c

∂x

∂a0∂y

∂a

∂x

∂b

∂y− ∂c

∂x

∂a0∂y

∂a

∂y

∂b

∂x− ∂c

∂y

∂a0∂x

∂a

∂x

∂b

∂y+∂c

∂y

∂a0∂x

∂a

∂y

∂b

∂x=

[∂a

∂x

∂a0∂y

∂b

∂x

∂c

∂y− ∂b

∂x

∂a0∂y

∂c

∂y

∂a

∂x] + [

∂a

∂x

∂a0∂y

∂b

∂y

∂c

∂x− ∂c

∂x

∂a0∂y

∂a

∂x

∂b

∂y] +

[∂a

∂y

∂a0∂x

∂b

∂x

∂c

∂y− ∂c

∂y

∂a0∂x

∂a

∂y

∂b

∂x] + [

∂a

∂y

∂a0∂x

∂b

∂y

∂c

∂x− ∂b

∂y

∂a0∂x

∂c

∂x

∂a

∂y] +

[∂b

∂x

∂a0∂y

∂c

∂x

∂a

∂y− ∂c

∂x

∂a0∂y

∂a

∂y

∂b

∂x] + [

∂b

∂y

∂a0∂x

∂c

∂y

∂a

∂x− ∂c

∂y

∂a0∂x

∂a

∂x

∂b

∂y]

= 0

• s A = k[x1, x2, x3]. P♦s♦♥s h := a0.

♦rs ♣♦r t♦t f, g, k ∈ A, ♦♥

f, h0 = (∂f

∂x1

∂h

∂x2− ∂f

∂x2

∂h

∂x1)x1, x20 + (

∂f

∂x1

∂h

∂x3− ∂f

∂x3

∂h

∂x1)x1, x30 +

(∂f

∂x2

∂h

∂x3− ∂f

∂x3

∂h

∂x2)x2, x30

g, k0 = (∂g

∂x1

∂k

∂x2− ∂g

∂x2

∂k

∂x1)x1, x20 + (

∂g

∂x1

∂k

∂x3− ∂g

∂x3

∂k

∂x1)x1, x30 +

(∂g

∂x2

∂k

∂x3− ∂g

∂x3

∂k

∂x2)x2, x30

f, h0g, k0 = (∂f

∂x1

∂h

∂x2− ∂f

∂x2

∂h

∂x1)(∂g

∂x1

∂k

∂x2− ∂g

∂x2

∂k

∂x1)(x1, x20)2

+(∂f

∂x1

∂h

∂x3− ∂f

∂x3

∂h

∂x1)(∂g

∂x1

∂k

∂x3− ∂g

∂x3

∂k

∂x1)(x1, x30)2+

(∂f

∂x2

∂h

∂x3− ∂f

∂x3

∂h

∂x2)(∂g

∂x2

∂k

∂x3− ∂g

∂x3

∂k

∂x2)(x2, x30)2

[(∂f

∂x1

∂h

∂x2− ∂f

∂x2

∂h

∂x1)(∂g

∂x1

∂k

∂x3− ∂g

∂x3

∂k

∂x1)+(

∂f

∂x1

∂h

∂x3− ∂f

∂x3

∂h

∂x1)(∂g

∂x1

∂k

∂x2−

∂g

∂x2

∂k

∂x1)](x1, x20)(x1, x30)

[(∂f

∂x1

∂h

∂x2− ∂f

∂x2

∂h

∂x1)(∂g

∂x2

∂k

∂x3− ∂g

∂x3

∂k

∂x2)+(

∂f

∂x2

∂h

∂x3− ∂f

∂x3

∂h

∂x2)(∂g

∂x1

∂k

∂x2−

∂g

∂x2

∂k

∂x1)](x1, x20)(x2, x30)+

[(∂f

∂x1

∂h

∂x3− ∂f

∂x3

∂h

∂x1)(∂g

∂x2

∂k

∂x3− ∂g

∂x3

∂k

∂x2)+(

∂g

∂x1

∂k

∂x3− ∂g

∂x3

∂k

∂x1)(∂f

∂x2

∂h

∂x3−

∂f

∂x3

∂h

∂x2)](x1, x30x2, x30)

♥ é♦♣♣♥t tt ①♣rss♦♥ ♦♥ ♦t♥t

♦♥t (x1, x20)2 st∂f

∂x1

∂h

∂x2

∂g

∂x1

∂k

∂x2− ∂f

∂x1

∂h

∂x2

∂g

∂x2

∂k

∂x1− ∂f

∂x2

∂h

∂x1

∂g

∂x1

∂k

∂x2+

∂f

∂x2

∂h

∂x1

∂g

∂x2

∂k

∂x1

♦é♥t (x1, x30)2 st∂f

∂x1

∂h

∂x3

∂g

∂x1

∂k

∂x3− ∂f

∂x1

∂h

∂x3

∂g

∂x3

∂k

∂x1− ∂f

∂x3

∂h

∂x1

∂g

∂x1

∂k

∂x3+

∂f

∂x3

∂h

∂x1

∂g

∂x3

∂k

∂x1

♦♥t (x2, x30)2 st∂f

∂x2

∂h

∂x3

∂g

∂x2

∂k

∂x3− ∂f

∂x2

∂h

∂x3

∂g

∂x3

∂k

∂x2− ∂f

∂x3

∂h

∂x2

∂g

∂x2

∂k

∂x3+

∂f

∂x3

∂h

∂x2

∂g

∂x3

∂k

∂x2

♦♥t x1, x2x1, x3 st∂f

∂x1

∂h

∂x3

∂g

∂x1

∂k

∂x2− ∂f

∂x1

∂h

∂x3

∂g

∂x2

∂k

∂x1− ∂f

∂x3

∂h

∂x1

∂g

∂x1

∂k

∂x2+∂f

∂x3

∂h

∂x1

∂g

∂x2

∂k

∂x1+

∂f

∂x1

∂h

∂x2

∂g

∂x1

∂k

∂x3− ∂f

∂x1

∂h

∂x2

∂g

∂x3

∂k

∂x1− ∂f

∂x2

∂h

∂x1

∂g

∂x1

∂k

∂x3+

∂f

∂x2

∂h

∂x1

∂g

∂x3

∂k

∂x1

♦♥t x1, x3x2, x3 st∂f

∂x1

∂h

∂x3

∂g

∂x2

∂k

∂x3− ∂f

∂x1

∂h

∂x3

∂g

∂x3

∂k

∂x2− ∂f

∂x3

∂h

∂x1

∂g

∂x2

∂k

∂x3+∂f

∂x3

∂h

∂x1

∂g

∂x3

∂k

∂x2+

∂g

∂x1

∂k

∂x3

∂f

∂x2

∂h

∂x3− ∂g

∂x1

∂k

∂x3

∂f

∂x3

∂h

∂x2− ∂g

∂x3

∂k

∂x1

∂f

∂x2

∂h

∂x3+

∂g

∂x3

∂k

∂x1

∂f

∂x3

∂h

∂x2

t♦♥ ♣r♠tt♦♥ (fgk) sr ♦♥t x1, x3x2, x3 ♥s

♥♦s ♦♥♥

∂f

∂x1

∂h

∂x3

∂g

∂x2

∂k

∂x3− ∂f

∂x1

∂h

∂x3

∂g

∂x3

∂k

∂x2− ∂f

∂x3

∂h

∂x1

∂g

∂x2

∂k

∂x3+∂f

∂x3

∂h

∂x1

∂g

∂x3

∂k

∂x2+

∂g

∂x1

∂k

∂x3

∂f

∂x2

∂h

∂x3− ∂g

∂x1

∂k

∂x3

∂f

∂x3

∂h

∂x2− ∂g

∂x3

∂k

∂x1

∂f

∂x2

∂h

∂x3+∂g

∂x3

∂k

∂x1

∂f

∂x3

∂h

∂x2+

∂g

∂x1

∂h

∂x3

∂k

∂x2

∂f

∂x3− ∂g

∂x1

∂h

∂x3

∂k

∂x3

∂f

∂x2− ∂g

∂x3

∂h

∂x1

∂k

∂x2

∂f

∂x3+

∂g

∂x3

∂h

∂x1

∂k

∂x3

∂f

∂x2+

∂k

∂x1

∂f

∂x3

∂g

∂x2

∂h

∂x3− ∂k

∂x1

∂f

∂x3

∂g

∂x3

∂h

∂x2− ∂k

∂x3

∂f

∂x1

∂g

∂x2

∂h

∂x3+∂k

∂x3

∂f

∂x1

∂g

∂x3

∂h

∂x2+

∂k

∂x1

∂h

∂x3

∂f

∂x2

∂g

∂x3− ∂k

∂x1

∂h

∂x3

∂f

∂x3

∂g

∂x2− ∂k

∂x3

∂h

∂x1

∂f

∂x2

∂g

∂x3+∂k

∂x3

∂h

∂x1

∂f

∂x3

∂g

∂x2+

∂f

∂x1

∂g

∂x3

∂k

∂x2

∂h

∂x3− ∂f

∂x1

∂g

∂x3

∂k

∂x3

∂h

∂x2− ∂f

∂x3

∂g

∂x1

∂k

∂x2

∂h

∂x3+

∂f

∂x3

∂g

∂x1

∂k

∂x3

∂h

∂x2

♥ ér q r♥r st ♥

♠ê♠ ♦♥ ♠♦♥tr q s ♦♥ts ♥s

x1, x2x1, x3, (x2, x30)2, (x1, x30)2, (x1, x20)2

s♦♥t t♦s ♥s ♦ù réstt

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦t ♦r r♣♣♦♥s q s G : Ep → F st ♥ ♣♣t♦♥ ♥ér ♥

ts②♠étrq t q ♣♦r t♦t y ∈ E ♣♣t♦♥ ♣rt Gy : En−1 → F st

♥ ♦rs G = 0.

♦t x ∈ L, ♥♦s é♥ss♦♥s s ♣♣t♦♥s ♥érs dx : Ltq(L,P ) → Ltq(L,P )♣r

(dxf)(x1, ..., xq) = ρ(x)f(x1, ..., xq)−q∑

i=1

f(x1, ..., [xi, x], ..., xq)

t Fx : Ltq+1(L,P ) → Ltq(L,P ) é♥ ♣r

(Fx(f))(x1, ..., xq) = f(x, x1, ..., xq).

s ♣♣t♦♥s s♦♥t és ♣r s rt♦♥s

Fy(dxf) = dx(Fy(f))− F[x,y](f)

t

Fx(dρf) = dxf − dρ(Fx(f)).

rt♦♥ ♥t ♥ ♣♣t♦♥ A♥ér d : L → End(Ltq(L,P )) é♥♣r x 7→ dx.

♦♥tr♦♥s q ♣♦r t♦t q ∈ N, d st ♥ r♣rés♥tt♦♥ L Lti(L,P ) ♦s♦♥s tr ♥ ♣r ♣r ♥t♦♥ sr q.

ρ(y)ρ(x)f − ρ(x)ρ(y)f − ρ([x, y])f = 0 ♣♦r t♦t f ∈ P t x, y ∈ L. ♠♦♥tr

q d : L → End(Lt0(L,P )) st ♥ ♠♦r♣s♠ èrs ♣♣♦s♦♥s

②♣♦tès r ♣♦r t♦t 1 ≦ k ≦ q− 1 t s♦t f ∈ Ltq(L,P )). P♦r t♦t z ∈ L,

♦♥

Fz(dydxf) = dy[Fz(dxf)]− F[z,y](dxf)

= dy(dxFz(f)− F[z,x](f)

)− F[z,y](dxf)

= dydxFz(f)− dy(F[z,x](f))

= dydxFz(f)− F[z,x](dyf)− F[[z,x],y](f)− F[z,y](dxf).

♦♥

Fz(dydxf)− Fz(dxdyf) = dydxFz(f)− dxdyFz(f) + (F[[z,x],y] + F[[z,y],x])(f)

= d[x,y]Fz(f)− F[[y,x],z](f)

= (d[x,y]Fz − F[[y,x],z])(f))

= Fz(d[x,y]).

Psq z ∈ L st rtrr ♦♥ ♦♥t ♣rès ♣r♥♣ ♥t♦♥ q d st

♥ ♥ r♣rés♥tt♦♥ L ♣r Ltq(L,P )) ♣♦r t♦t q.♦♥tr♦♥s q r♠♠ s♥t st ♦♠♠tt ♣♦r t♦t q ∈ N t x ∈ L

Ltq(L,P )) dρ //

dx

Ltq+1(L,P ))

dx

Ltq(L,P ))dρ

// Ltq+1(L,P ))

f ∈ Lt0(L,P )) = P, ♦rs ♣♦r t♦t y ∈ L ♦♥

(dxdρf)(y) = ρ(x)(dρf)(y)− (dρf)([y, x])

= (ρ(x)ρ(y)− ρ[y, x])(f)

= ρ(y)ρ(x)(f) = ρ(y)(dxf) = (dρdxf)(x)

♣♣♦s♦♥s q r♠♠ st ♦♠♠tt ♣♦r t♦t 1 ≦ k ≦ q − 1 t s♦t

f ∈ Ltq+1(L,P )); q > 0. ♥ ♣♣q♥t s rt♦♥s t t q d st

♥ ♠♦r♣s♠ èrs ♦♥ ♦t♥t

Fy(dρdxf)− Fy(dxdρf)

= dydxf − dρ[Fy(dxf)]− dx[Fy(dρf)] + F[y,x](dρf)

= dydxf − dρdxFy(f)− dρ(F[y,x])− dx[Fy(dρf)] + d[y,x]f − dρ(F[y,z](f))

= dydxf − dρdx(Fy(f))− dxdyf + dxdρ(Fy(f)) + d[y,x]f

= dxdρ(Fy(f))− dρdx(Fy(f)) = 0.

s ♣r♦♣rétés ♥♦s é♠♦♥tr♦♥s ♣r♦♣♦st♦♥ ♣r ♥t♦♥ sr ♦rr

s ♥s ♣rès é ♣r Pr♦♣♦st♦♥ réstt st r

♣♦r q = 0, 1. ♣♣♦s♦♥s réstt r ♣♦r f ∈ Ltk(L,P )) 1 ≦ k ≦ q − 1

t s♦t f ∈ Ltq(L,P )), q > 0. ♣rès q ♣réè ♦♥

Fx(dρdρf) = dxdρf − dρ[Fx((dρf))] = dxdρf − dρdxf + dρdρ(Fx(f)) = 0

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦t a, b ∈ A t u, v ∈ S. s ♣r♦♣rétés Pr♦♣♦st♦♥ ♥♦s és♦♥s q

LH[a

d(u)

u]

(d(v)

v)

= aLH[d(u)

u]

(d(v)

v) + σ[H(

d(u)

u)]

(d(v)

v

)d(a)

= aLH[d(u)

u]

(d(v)

v) +

1

uσ(H d(u))(d(v)

v)d(a)

= aLH[d(u)

u]

(d(v)

v) +

1

uσ(u,−)(d(v)

v)d(a)

= ad( 1uvu, v) +

1

uvu, vd(a).

♣rès Pr♦♣♦st♦♥ ♦♥

LH[d(u)

u]

(bd(v)

v)

= [H(d(u)

u)](b)

d(u)

u+ bL

H[d(u)

u]

(d(v)

v)

=1

uu, bd(v)

v+ bd(

1

uvu, v)

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

♣rt♥t

LH(a

d(u)

u)

(bd(v)

v)

= aLH[d(u)

u]

(bd(v)

v) + σ(H(

d(u)

u)))

(bd(v)

v

)d(a)

=a

uu, bd(v)

v+

b

uvu, vd(a) + abd(

1

uvu, v)

♥ ♥trrtss♥t s rôs u t v ♥♦s ♦t♥♦♥s

LH(b

d(v)

v)

(ad(u)

u) =

b

vv, ad(u)

u+

a

uvv, ud(b) + abd(

1

uvv, u)

Psq ω0(x, y) := [Φ(x)]y ♣♦r t♦t x, y ∈ ΩA(log I)ω(a

d(u)

u, bd(v)

v) =

ab

uvu, v

♦rs

dω(ad(u)

u, bd(v)

v) = d[

ab

uvu, v] = abd[

1

uvu, v] + d(ab).(

1

uvu, v)

= abd[1

uvu, v] + bd(a).(

1

uvu, v) + ad(b).(

1

uvu, v)

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦♥t a ∈ A t u, v, w ∈ S. ♣rès ♦r♦r ♦♥ [du

u,dv

v]ω =

d(1

uvu, v) t [da, du

u]ω = d(

1

ua, u).

t♥t ♦♥♥é q strtr P♦ss♦♥ −,− st ♦rt♠q ♣r♥♣ ♦♥

I, ♦♥ 1

uvu, v ∈ A. ♥ ♦♥

[[du

u,dv

v

]

ω

,dw

w

]

ω

=

[d(

1

uvu, v), dw

w

]

ω

= d(1

w 1

uvu, v, w).

[[dv

v,dw

w

]

ω

,du

u

]

ω

=

[d(

1

vwv, w), du

u

]

ω[[dw

w,du

u

]

ω

,dv

v

]

ω

=

[d(

1

uww, u), dv

v

]

ω

.

r ♥ ♣♣q♥t ♠♠ ♦♥ ♦t♥t

1

w 1

uvu, v, w =

1

w(1

uvu, v, w − 1

u2v2u, vuv,w)

=1

uvwu, v, w − 1

wu2vu, vu,w − 1

wuv2u, vv, w

1

u 1

vwv, w, u =

1

u(1

vwv, w, u − 1

v2w2v, wvw, u)

=1

vwuv, w, u − 1

uv2wv, wv, u − 1

uvw2v, ww, u

1

v 1

wuw, u, v =

1

v(1

vuw, u, v − 1

w2u2w, uwu, v)

=1

wuvw, u, v − 1

vvw2uw, uw, v − 1

vwu2w, uu, v.

♥ ♦t♥t ♦♥[[du

u,dv

v

]

ω

,dw

w

]

ω

+

[[dv

v,dw

w

]

ω

,du

u

]

ω

+

[[dw

w,du

u

]

ω

,dv

v

]

ω

=1

uvwu, v, w − 1

wu2vu, vu,w − 1

wuv2u, vv, w

+1

vwuv, w, u − 1

uv2wv, wv, u − 1

uvw2v, ww, u

+1

wuvw, u, v − 1

vvw2uw, uw, v − 1

vwu2w, uu, v

=1

uvw(v, w, u+ w, u, v+ u, v, w)

= 0

r♥èr été é♦ ♥tté ♦ −,−.

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

P♦r q st ♣r♠èr ssrt♦♥ ét♥t ♦♥♥és u, v ∈ S t w ∈ A ♥♦s

♦♥s ♥tté s♥t [[du

u,dv

v

], dw

]=

[d(

1

uvu, v), dw

]= d

( 1

uvu, v, w

).

r

1

uvu, v, w

=

1

uvu, v, w − 1

uv2u, vv, w − 1

vu2u, vu,w

♣r ♦♥séq♥t[[du

u,dv

v

], dw

]= d

(1

uvu, v, w − 1

uv2u, vv, w − 1

vu2u, vu,w

).

tts ♠t♥s ♦♥ [[dv

v, dw

],du

u

]=

[d

(1

vv, w

),du

u

]= d

(1

u

1

vv, w, u

).

Psq1

u

1

vv, w, u

=

1

u

(1

vv, w, u − 1

v2v, wv, u

)=

1

uvv, w, u−

1

uv2v, wv, u

♦♥

[[dv

v, dw

],du

u

]= d

(1

uvv, w, u − 1

uv2v, wv, u

)

t ♦♥[[dw,

du

u

],dv

v

]=

[(1

uw, u

),dv

v

]= d

(1

v

1

uw, u, v

).

Pr rs1

v

1

uw, u, v

=

1

vuw, u, v − 1

vu2w, uu, v.

♦♥

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

[[dw,

du

u

],dv

v

]= d

(1

vuw, u, v − 1

vu2w, uu, v

).

♥tté ♦ −,− ♦♥t ① s sss ♦♥♥ [[du

u,dv

v

], dw

]+

[[dv

v, dw

],du

u

]+

[[dw,

du

u

],dv

v

]= 0.

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦♥t a1, a2, a3, u1, u2 t u3 é♥s ♣r s ②♣♦tèss Pr♦♣♦st♦♥

P ♣rès ♠♠ ♥♦s ♦♥s 1

u3 1

u1u2u1, u2, u3+

1

u1 1

u2u3u2, u3, u1+

1

u2 1

u3u1u3, u1, u2

=1

u1u2u3u1, u2, u3 −

1

u3u1u22u1, u2u2, u3 −

1

u3u21u2u1, u2u1, u3+

+1

u1u2u3u2, u3, u1 −

1

u3u1u22u2, u3u2, u1 −

1

u23u1u2u2, u3u3, u1+

1

u1u2u3u3, u1, u2 −

1

u3u21u2u3, u1u1, u2 −

1

u23u1u2u3, u1u3, u2

=1

u1u2u3(u1, u2, u3+ u2, u3, u1+ u3, u1, u2)+

−u2, u3u3u1u22

(u1, u2+ u2, u1)−u1, u2u3u21u2

(u1, u3+ u3, u1)+

−u3, u1u23u1u2

(u2, u3+ u3, u2)♣♥♥t r♦t −,− ét♥t ♥ts②♠étrq ui, uj + uj , ui = 0

♣♦r t♦t (i, j).

♥ tr♠♥ ♣r P ♥ ts♥t ♥tté ♦ −,−.

P ♥ ♣♣q♥t ♠♠ ♦♥ ♦t♥t a1u1

a2u2

u2, a3, u1du3u3

=

(a1u1u2

a2u2, a3, u1 −a1a2u1u22

u2, a3u2, u1)du3u3

=

(a1a2u1u2

u2, a3, u1+a1u1u2

u2, a3a2, u1 −a1a2u1u22

u2, a3u2, u1)du3u3

.

Pr ♥ rs♦♥♥♠♥t ♥♦ ♦♥ é♠♦♥tr s trs ♣r♦♣rétés

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♣rès Pr♦♣♦st♦♥ ♦♥

[[a1du1u1

, a2du2u2

], a3

du3u3

]+

[[a2du2u2

, a3du3u3

], a1

du1u1

]+

[[a3du3u3

, a1du1u1

], a2

du2u2

]=

a1u1u2

u1, a2u2, a3du3u3

+a3a1u3u1

u1, a2, u3du2u2

+a3u3u1

u1, a2a1, u3du2u2

+

−a3a1u21u3

u1, a2u1, u3du2u2

+a1a3u1

u1, a2d(1

u2u3u2, u3) +

a2u2u1

a1, u2u1, a3du3u3

+

+a3a2u3u2

a1, u2, u3du1u1

+a3u3u2

a1, u2a2, u3du1u1

− a3a2u3u22

a1, u2u2, u3du1u1

+

a2a3u2

a1, u2d(1

u1u3u1, u3) +

a1a2u1u2

u1, u2, a3du3u3

− a1a2u1u22

u1, u2u2, a3du3u3

+

−a1a2u21u2

u1, u2u1, a3du3u3

+a3a1u3

a2, u3d(1

u1u2u1, u2) +

a3a2u3

a1, u3d(1

u1u2u1, u2)

+a1a2a3d(1

u3 1

u1u2u1, u2, u3)

a2u2u3

u2, a3u3, a1du1u1

+a1a2u1u2

u2, a3, u1du3u3

+a1u1u2

u2, a3a2, u1du3u3

+

−a1a2u22u1

u2, a3u2, u1du3u3

+a2a1u2

u2, a3d(1

u3u1u3, u1) +

a3u3u2

a2, u3u2, a1du1u1

+

+a1a3u1u3

a2, u3, u1du2u2

+a1u1u3

a2, u3a3, u1du2u2

− a1a3u1u23

a2, u3u3, u1du2u2

+

a3a1u3

a2, u3d(1

u2u1u2, u1) +

a2a3u2u3

u2, u3, a1du1u1

− a2a3u2u23

u2, u3u3, a1du1u1

+

−a2a3u22u3

u2, u3u2, a1du1u1

+a1a2u1

a3, u1d(1

u2u3u2, u3) +

a1a3u1

a2, u1d(1

u2u3u2, u3)

+a2a3a1d(1

u1 1

u2u3u2, u3, u1)

a3u3u1

u3, a1u1, a2du2u2

+a2a3u2u3

u3, a1, u2du1u1

+a2u2u3

u3, a1a3, u2du1u1

+

−a2a3u23u2

u3, a1u3, u2du1u1

+a3a2u3

u3, a1d(1

u1u2u1, u2) +

a1u1u3

a3, u1u3, a2du2u2

+

+a2a1u2u1

a3, u1, u2du3u3

+a2u2u1

a3, u1a1, u2du3u3

− a2a1u2u21

a3, u1u1, u2du3u3

+

a1a2u1

a3, u1d(1

u3u2u3, u2) +

a3a1u3u1

u3, u1, a2du2u2

− a3a1u3u21

u3, u1u1, a2du2u2

+

−a3a1u23u1

u3, u1u3, a2du2u2

+a2a3u2

a1, u2d(1

u3u1u3, u1) +

a2a1u2

a3, u2d(1

u3u1u3, u1)

+a3a1a2d(1

u2 1

u3u1u3, u1, u2).

♥ rt s s ♥térrs été sss st éq♥t à

[[a1du1u1

, a2du2u2

], a3

du3u3

]+

[[a2du2u2

, a3du3u3

], a1

du1u1

]+

[[a3du3u3

, a1du1u1

], a2

du2u2

]=

a1u1u2

u2, a3 (u1, a2+ u2, a1)du3u3

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

a3a1u3u1

(u1, a2, u3+ a2, u3, u1+ u3, u1, a2)du2u2

+u1, a2a3u3u1

(a1, u3+ u3, a1)du2u2

+−u1, a2a3a1u21u3

(u1, u3+ u3, u1)du2u2

+

a1a3u1

(u1, a2+ a2, u1) d(1

u2u3u2, u3) + a1, u2

a2u2u1

(u1, a3+ a3, u1)du3u3

+a3a2u3u2

(a1, u2, u3+ u2, u3, a1+ u3, a1, u2)du1u1

+a2, u3a3u3u2

(a1, u2+ u2, a1+)du1u1

− u2, u3a3a2u3u22

(a1, u2+ u2, a1)du1u1

+

a2a3u2

a1, u2d(

1

u1u3u1, u3+

1

u3u1u3, u1

)

+a1a2u1u2

(u1, u2, a3+ u2, a3, u1+ a3, u1, u2)du3u3

+

−u2, a3a1a2u1u22

(u1, u2+ u2, u1)du3u3

− u1, u2a1a2u21u2

(u1, a3+ a3, u1)du3u3

+a3a1u3

a2, u3d(1

u1u2(u1, u2+ u2, u1))) +

a3a2u3

d(1

u1u2u1, u2 (a1, u3+ u3, a1))+

u3, a1a2u2u3

(u2, a3+ a3, u2)du1u1

+a2a1u2

(u2, a3+ a3, u2) d(1

u3u1u3, u1)+

a3, u1a1u1u3

(a2, u3+ u3, a2)du2u2

− u3, u1a1a3u1u23

(a2, u3+ u3, a2)du2u2

−u3, a1a2a3u2u23

(u2, u3+ u3, u2)du1u1

+a1a2u1

a3, u1d(1

u2u3(u2, u3) + u3, u2))

+a1a2a3d

(1

u3 1

u1u2u1, u2, u3+

1

u1 1

u2u3u2, u3, u1+

1

u2 1

u3u1u3, u1, u2

)= 0.

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

é♦ s ♣r♦♣rétés s strtrs P♦ss♦♥ ♦rt♠qs ♣r♥♣s

étés à Pr♦♣♦st♦♥ q ♦♥ s étés q s♥t

[[a1

du1u1

, a2du2u2

]ω, b3dv3

]

ω

=a1u1u2

u1, a2u2, b3dv3 +b3a1u1

u1, a2, v3du2u2

+

b3u1

u1, a2a1, v3du2u2

− b3a1u21

u1, a2u1, v3du2u2

+a1b3u1

u1, a2d(1

u2u2, v3)+

a2u2u1

a1, u2u1, b3dv3 +b3a2u2

a1, u2, v3du1u1

+b3u2

a1, u2a2, , v3du1u1

+

−b3a2u22

a1, u2u2, v3du1u1

+a2b3u2

a1, u2d(1

u1u1, v3) +

a2a1u1u2

u1, u2, b3dv3+

−a1a2u21u2

u1, u2u1, b3dv3 −a1a2u1u22

u1, u2u2, b3dv3 + b3a1a2, v3d(1

u1u2u1, u2)

+b3a2a1, v3d(1

u1u2u1, u2) + a1a2b3d(

1

u1u2u1, u2, v3).

[[a2

du2u2

, b3dv3]ω, a1du1u1

]

ω

=a2u2

u2, b3v3, a1du1u1

+a1a2u1u2

u2, b3, u1dv3

+a1u1u2

u2, b3a2, u1dv3 −a1a2u1u22

u2, b3u2, u1dv3 +a1a2u2

u2, b3d(1

u1v3, u1)

+b3u2

a2, v3u2, a1du1u1

+a1b3u1

a2, v3, u1du2u2

+a1u1

a2, v3b3, u2du2u2

+a1b3a2, v3d(1

u1u2u2, u1) +

a2b3u2

u2, v3, a1du1u1

− a2b3u22

u2, a1u2, v3du1u1

+a1a2u1

b3, u1d(1

u2u2, v3) +

a1b3u1

a2, u1d(1

u2u2, v3)+

a2b3a1d(1

u1 1

u2u2, v3, u1)

t[[b3dv3, a1

du1u1

]ω, a2du2u2

]

ω

=b3u1

v3, a1u1, a2du2u2

+a2b3u2

v3, a1, u2du1u1

+a2u2

v3, a1b3, u2du1u1

+ b3a2v3, a1d(1

u1u2u1, u2) +

a1u1

b3, u1v3, a2du2u2

+a2a1u1u2

b3, u1, u2dv3 +a2u1u2

b3, u1a1, u2dv3 −a2a1u21u2

b3, u1u1, u2dv3+a1a2u1

b3, u1d(1

u2v3, u2) +

b3a1u1

v3, u1, a2du2u2

− b3a1u21

v3, u3u1, a2du2u2

+a1a2u2

b3, u2d(1

u1v3, u1) +

a2b3u2

a1, u2d(1

u1v3, u1)+

a1b3a2d(1

u2 1

u1v3, u1, u2).

r ♠♠r r♦t r♥èr été ♣t sérr s♦s ♦r♠

u2, b3a1u1u2

(u1, a2+ a2, u1) dv3 +b3u1

u1, a2 (a1, v3+ v3, a1)du2u2

+

b3a1u1

(u1, a2, v3+ v3, u1, a2+ a2, v3, u1+)du2u2

+

−u1, a2b3a1u21

(u1, v3+ v3, u1)du2u2

+a1b3u1

(u1, a2+ a2, u1) d(1

u2u2, v3)+

a2u2u1

a1, u2(u1, b3+ b3, u1)dv3 + a2, v3b3u2

(a1, u2+ u2, a1)du1u1

+

b3a2u2

(a1, u2, v3+ u2, v3, a1+ v3, a1, u2)du1u1

+

−u2, v3b3a2u22

(a1, u2+ u2, a1)du1u1

+a2b3u2

a1, u2d(1

u1(u1, v3+ v3, u1))+

a2a1u1u2

(u1, u2, b3+ u2, b3, u1+ b3, u1, u2)dv3+

−u1, u2a1a2u21u2

(u1, b3+ b3, u1)dv3 − u2, b3a1a2u1u22

(u1, u2+ u2, u1dv3)dv3

+b3a1a2, v3d(1

u1u2(u2, u1+ u1, u2)) + b3a2(a1, v3+ v3, a1)d(

1

u1u2u1, u2)

+a1a2b3d(1

u1u2u1, u2, v3+

1

u1 1

u2u2, v3, u1+

1

u2 1

u1v3, u1, u2)+

v3, a1a2u2

(u2, b3+ b3, u2)du1u1

+a1a2u2

(u2, b3+ b3, u2)d(1

u1v3, u1)

+b3, u2a1u1

(a2, v3+ v3, a2)du2u2

++b3, u1a1a2u1

d(1

u2(u2, v3+ v3, u2)).

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

è é♠♦♥strt♦♥

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦♥t u1, u3 ∈ S a1, a3, b2 t v2 ♥s A ♣rès r♠rq ♦s ♦♥s [[a1

du1u1

, b2dv2]ω, a3du3u3

]

ω

+

[[b2dv2, a3

du3u3

]ω, a1du1u1

]

ω

+

[[a3

du3u3

, a1du1u1

]ω, b2dv2

]

ω

=

a1u1

u1, b2v2, a3du3u3

+a3a1u3u1

u1, b2, u3dv3 +a3u3u1

a1, u3u1, b2dv3+

−a3a1u3u21

u1, b2u1, u3dv2 +a1a3u1

u1, b2d(1

u3v2, u3) +

b2u1

a1, v2u1, a3du3u3

+

a3b2u3

a1, v2, u3du1u1

+a3u3

b2, u3a1, v2du1u1

+ a3b2a1, v2d(1

u1u3u1, u3)

+a1b2u1

u1, v2, a3du3u3

− a1b2u21

u1, a3u1, v2du3u3

+a3a1u3

b2, u3d(1

u1u1, v2)

+a3b2u3

a1, u3d(1

u1u1, v2) + a1b2a3d(

1

u3 1

u1u1, v2, u3)

+b2u3

v2, a3u3, a1du1u1

+a1b2u1

v2, a3, u1du3u3

+a1u1

v2, a3b2, u1du3u3

+ b2a1v2, a3d(1

u3u1u3, u1) +

a3u3

b2, u3v2, a1du1u1

+a1a3u3u1

b2, u3, u1dv2 +a1u3u1

b2, u3a3, u1dv2 −a1a3u23u1

b2, u3u3, u1dv2+a3a1u3

b3, u3d(1

u1v2, u1) +

b2a3u3

v2, u3, a1du1u1

− b2a3u23

v2, u3u3, a1du1u1

+a3a1u1

b2, u1d(1

u3v2, u3) +

a1b2u1

a3, u1d(1

u3v2, u3)+

a3b3a1d(1

u1 1

u3v2, u3, u1)

+a3u3u1

u3, a1u1, b2dv2 +b2a3u3

u3, a1, v2du1u1

+

b2u3

u3, a1a3, v2du1u1

− b2a3u23

u3, a1u3, v2du1u1

+a3b2u3

u3, a1d(1

u1u1, v2)+

a1u1u3

a3, u1u3, b2dv2 +b2a1u1

a3, u1, v2du3u3

+b2u1

a3, u1a1, v2du3u3

+

−b2a1u21

a3, u1u1, v2du3u3

+a1b2u1

a3, u1d(1

u3u3, v2) +

a1a3u3u1

u3, u1, b2dv2+

−a3a1u23u1

u3, u1u3, b2dv2 −a3a1u3u21

u3, u1u1, b2dv2 + b2a3a1, v2d(1

u3u1u3, u1)

+b2a1a3, v2d(1

u3u1u3, u1) + a3a1b2d(

1

u3u1u3, u1, v2).

♥ ♥s

[[a1

du1u1

, b2dv2]ω, a3du3u3

]

ω

+

[[b2dv2, a3

du3u3

]ω, a1du1u1

]

ω

+

[[a3

du3u3

, a1du1u1

]ω, b2dv2

]

ω

= v2, a3a1u1

(u1, b2+ b2, u1)du3u3

+ u1, b2a3u3u1

(a1, u3+ u3, a1)dv2a3a1u3u1

(u1, b2, u3+ b2, u3, u1+ u3, u1, b2)dv2+

−u1, b2a3a1u3u21

(u1, u3+ u3, u1)dv2 +a1a3u1

(u1, b2+ b2, u1)d(1

u3v2, u3)+

a3b2u3

(a1, v2, u3+ u3, a1, v2+ v2, u3, a1)du1u1

+

+b2, u3a3u3

(a1, v2+ v2, a1)du1u1

+ a3b2a1, v2d(1

u1u3(u1, u3+ u3, u1)

+(a1b2u1

u1, v2, a3+ v2, a3, u1+ a3, u1, v2)du3u3

−u1, v2a1b2u21

(u1, a3+ a3, u1)du3u3

+a3a1u3

b2, u3d(1

u1(u1, v2+ v2, u1))+

a3b2u3

(a1, u3+ u3, a1)d(1

u1u1, v2) +

b2u3

v2, a3(u3, a1+ a1, u3)du1u1

+b2a1(v2, a3+ a3, v2)d(1

u3u1u3, u1) + a3, u1

a1u3u1

(b2, u3+ u3, b2dv2

−a1a3u23u1

b2, u3(u3, u1+ u1, u3)dv2 −b2a3u23

u3, a1(v2, u3+ v2, u3)du1u1

.

é♠♦♥strt♦♥ é♦rè♠

♦t αi =dh

h+ α1

i , ♦♥ (LH(α1)

α2|H(α3))

=

(1

hLH(dh)α

12|1

hH(dh)

)−

(H(dh)(α1

2)

h

dh

h| 1hH(dh)

)+

(1

hLH(dh)α

12|H(α1

3)

)−

(H(dh)(α1

2)

h

dh

h|H(α1

3)

)−

(H(α1

1)h

h

dh

h| 1hH(dh)

)−

(H((α1

1)

h

dh

h|H(α1

3)

)+

(1

hLH(α1

1)dh|

1

hH(dh)

)+

(1

hLH(α1

1)dh|H(α1

3)

)+

(LH(α1

1)α

12|1

hH(dh)

)+(LH(α1

1)α

12|H(α1

3))

ér♦♥s r♥èr été s♦s ♦r♠ s♥t

(LH(α1)

α2|H(α3))

=1

h2(LH(dh)α

12|H(dh)

)+

1

h

(LH(dh)α

12|H(α1

3))

−H(dh)(α1

2)

h2(dh|H(α1

3))

− H(α11)h

h2(dh|H(α1

3))

+1

h2

(LH(α1

1)dh|H(dh)

)+

1

h

(LH(α1

1)dh|H(α1

3))+

1

h

(LH(α1

1)α

12|H(dh)

)+(LH(α1

1)α

12|H(α1

3)).

♥ s♦♠♠♥t s♦s ♣r♠tt♦♥ ②q ♦♥ ♦t♥t (LH(α1)

α2|H(α3))+ =

1

h2(LH(dh)α

12|H(dh)

)+

1

h

(LH(dh)α

12|H(α1

3))

−H(dh)α1

2

h2(dh|H(α1

3))

− H(α11)(h)

h2(dh|H(α1

3))

+1

h2

(LH(α1

1)dh|H(dh)

)+

1

h

(LH(α1

1)dh|H(α1

3))

+1

h

(LH(α1

1)α

12|H(dh)

)+

(LH(α1

1)α

12|H(α1

3))

+

1

h2(LH(dh)α

13|H(dh)

)+

1

h

(LH(dh)α

13|H(α1

1))

− H(dh)(α13)

h2(dh|H(α1

1))

−

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

H(α12)h

h2(dh|H(α1

1))

+1

h2

(LH(α1

2)dh|H(dh)

)+

1

h

(LH(α1

2)dh|H(α1

1))

+

1

h

(LH(α1

2)α

13|H(dh)

)+(LH(α1

2)α

13|H(α1

1))+

1

h2(LH(dh)α

11|H(dh)

)+1

h

(LH(dh)α

11|H(α1

2))−

H(dh)(α11)

h2(dh|H(α1

2))

− H(α13)h

h2(dh|H(α1

2))

+1

h2

(LH(α1

3)dh|H(dh)

)+

1

h

(LH(α1

3)dh|H(α1

2))+

1

h

(LH(α1

3)α

11|H(dh)

)+(LH(α1

3)α

11|H(α1

2))

♥ ♦t♥t (LH(α1)

α2|H(α3))+ =

[(LH(α1

3)α

11|H(α1

2))+(LH(α1

1)α

12|H(α1

3))+(LH(α1

2)α

13|H(α1

1))]

1

h

[(LH(α1

3)α

11|H(dh)

)+(LH(α1

1)dh|H(α1

3))+(LH(dh)α

13|H(α1

1))]

+

1

h

[(LH(α1

3)dh|H(α1

2))+(LH(dh)α

12|H(α1

3))+(LH(α1

2)α

13|H(dh)

)]+

1

h2

[(LH(α1

3)dh|H(dh)

)+(LH(dh)dh|H(α1

3))+(LH(dh)|α1

3H(dh))]

+

1

h2

[(LH(α1

3)dh|H(dh)

)+(LH(dh)dh|H(α1

3))+(LH(dh)α

13|H(dh)

)]+

1

h

[(LH(dh)α

11|H(α1

2))+(LH(α1

1)α

12|H(dh)

)+(LH(α1

2)dh|H(α1

1))]

+

− 1

h2[H(α1

1)(dh)(dh|H(α1

2))+H(α1

3)(dh)(dh|H(α1

2))]

+

− 1

h2[H(α1

3)(dh)(dh|H(α1

1))+H(α1

2)(dh)(dh|H(α1

1))]

+

− 1

h2[H(dh)(α1

2)(dh|H(α1

3))+H(dh)(α1

1)(dh|H(α1

3))]

Psq H st ♠t♦♥♥ s rt♦♥s s♦♥t ♥s ♥ ♦♥

(LH(α1)

α2|H(α3))+ =

−H(dh)(α11)

h2(dh|H(α1

2))− H(dh)(α1

3)

h2(dh|H(α1

2))

−H(dh)(α13)

h2(dh|H(α1

1))− H(dh)(α1

2)

h2(dh|H(α1

1))

−H(dh)(α12)

h2(dh|H(α1

3))− H(dh)(α1

1)

h2(dh|H(α1

3))

= −[H(α1

1)(dh)

h2(dh|H(α1

2))+H(dh)(α1

2)

h2(dh|H(α1

1))]

−[H(α1

3)(dh)

h2(dh|H(α1

2))+H(dh)(α1

2)

h2(dh|H(α1

3))]

−[H(α1

3)(dh)

h2(dh|H(α1

1))+H(dh)(α1

1)

h2(dh|H(α1

3))]

= 0.

♥ rt t qq s♦t αi = α0i

dh

h+ α1

i ∈ ΩX(logD) qq s♦t i ∈1, 2, 3 ♦♥

Lα01

hH(dh)+H(α1

1)

(α02

dh

h+ α1

2) = α01

H(dh)

h.(α0

2)dh

h

+α01

hLH(dh)α

12 +H(dh).α1

2

dα01

h− α0

1

H(dh)

h.α1

2

dh

h

+H(α11).(α

02)dh

h− α0

2

H(α11)

h.dh

dh

h+α02

hLH(α1

1)dh+ LH(α1

1)α

12

s♥ st q

(LH(α1)

α2|H(α3))=

(α01

hH(dh)(α0

2)dh

h|α

03

hH(dh)

)+

(α01

hH(dh)(α0

2)dh

h|H(α1

3)

)+

(α01

hLH(dh)α

12|α03

hH(dh)

)+

(α01

hLH(dh)α

12|H(α1

3)

)+

(H(dh).α1

2

dα01

h|α

03H(dh)

h

)+

(H(dh).α1

2

dα01

h|H(α1

3)

)+

(α01

H(dh).α12

h

dh

h|α0

3

H(dh)

h

)+

(α01

H(dh).α12

h

dh

h|H(α1

3)

)+

(H(α1

1)(α02)dh

h|α

03

hH(dh)

)+

(H(α1

1)(α02)dh

h|H(α1

3)

)−

(α02

H(α11).dh

h

dh

h|α

03

hH(dh)

)−

(α02

H(α11)

h.dh

dh

h|H(α1

3)

)+

(α02

hLH(α1

1)dh|

α03

hH(dh)

)+

(α02

hLH(α1

1)dh|H(α1

3)

)+

(LH(α1

1)α

12|α03

hH(dh)

)+(LH(α1

1)α

12|H(α1

3)).

rtèr ♥ts②♠étrq r♦t P♦ss♦♥ ♦♥ ét(LH(α1)

α2|H(α3))+ =

α01H(dh)(α0

2)

h2(dh|H(α1

3))+α01α

03

h2(LH(dh)α

12|H(dh)

)+

α01

h

(LH(dh)α

12|H(α1

3))+ α0

3

H(dh).α12

h2(dα0

1|H(dh))+H(dh).α1

2

h

(dα0

1|H(α13))+

α03α

02

h2

(LH(α1

1)dh|H(dh)

)− α0

1

H(dh).α12

h2(dh|H(α1

3))+H(α1

1)(α02)

h

(dh|H(α1

3))

−α02

H(α11).dh

h2(dh|H(α1

3))+α02

h

(LH(α1

1)dh|H(α1

3))+α03

h

(LH(α1

1)α

12|H(dh)

)+

(LH(α1

1)α

12|H(α1

3))+α02H(dh)(α0

3)

h2(dh|H(α1

1))+α02α

01

h2(LH(dh)α

13|H(dh)

)+

α02

h

(LH(dh)α

13|H(α1

1))+ α0

1

H(dh).α13

h2(dα0

2|H(dh))+H(dh).α1

3

h

(dα0

2|H(α11))+

α01α

03

h2

(LH(α1

2)dh|H(dh)

)− α0

2

H(dh).α13

h2(dh|H(α1

1))+H(α1

2)(α03)

h

(dh|H(α1

1))

−α03

H(α12).dh

h2(dh|H(α1

1))+α03

h

(LH(α1

2)dh|H(α1

1))+α01

h

(LH(α1

2)α

13|H(dh)

)+

(LH(α1

2)α

13|H(α1

1))+α02H(dh)(α0

3)

h2(dh|H(α1

1))+α02α

01

h2(LH(dh)α

13|H(dh)

)+

α03

h

(LH(dh)α

11|H(α1

2))+ α0

2

H(dh).α11

h2(dα0

3|H(dh))+H(dh).α1

1

h

(dα0

3|H(α12))+

α02α

01

h2

(LH(α1

3)dh|H(dh)

)− α0

3

H(dh).α11

h2(dh|H(α1

2))+H(α1

3)(α01)

h

(dh|H(α1

2))

−α01

H(α13).dh

h2(dh|H(α1

2))+α01

h

(LH(α1

3)dh|H(α1

2))+α02

h

(LH(α1

3)α

11|H(dh)

)+(

LH(α1

3)α

11|H(α1

2)).

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

♣rès rr♦♣♠♥t ♦♥ ♦t♥t

(LH(α1)

α2|H(α3))+ =

α01

h2[(H(dh)|d(α0

2)) (dh|H(α1

3))+(H(dh)|α1

3

) (dα0

2|H(dh))]

+

α01α

03

h2

[(LH(dh)α

12|H(dh)

)+(LH(α1

2)dh|H(dh)

)+

(LH(dh)dh|H(α1

2))]

+

α01

h

[(LH(dh)α

12|H(α1

3))+(LH(α1

2)α

13|H(dh)

)+(LH(α1

3)dh|H(α1

2))]

+

α03

h2[(H(dh)|α1

2

) (dα0

1|H(dh))+

(H(dh)|dα0

1

) (dh|H(α1

2))]

+

1

h

[(H(dh)|α1

2

) (dα0

1|H(α13))+(dh|H(α1

2)) (dα0

1|H(α13))]

+

α03α

02

h2

[(LH(α1

1)dh|H(dh)

)+(LH(dh)dh|H(α1

1))+

(LH(dh)α

11|H(dh)

)]+

− α01

h2[(H(dh)|α1

2

) (dh|H(α1

3))+(dh|H(α1

2)) (dh|H(α1

3))]

+

1

h

[(H(α1

1)|dα02

) (dh|H(α1

3))+(H(α1

1)|dα02

) (H(dh)|α1

3

)]+

− α02

h2[(H(α1

1)|dh) (dh|H(α1

3))+(H(α1

1)|dh) (H(dh)|α1

3

)]+

α02

h

[(LH(α1

1)dh|H(α1

3))+(LH(dh)α

13|H(α1

1))+(LH(α1

3)α

11|H(dh)

)]+

α03

h

[(LH(α1

1)α

12|H(dh)

)+(LH(α1

2)dh|H(α1

1))+(LH(dh)α

11|H(α1

2))]

(LH(α1

1)α

12|H(α1

3))+(LH(α1

2)α

13|H(α1

1))+(LH(α1

3)α

11|H(α1

2))

t q H stst s rt♦♥ ♥tr♥ ♥té s rt♦♥s

♥ rést q(LH(α1)

α2|H(α3))+ = 0

Pr rs ♣♦r t♦s α = α0dh

h+ α1 t β = β0

dh

h+ β1. ♦♥

(H(α)|β

)=

=

(α0H(dh)

h+H(α1)|β0

dh

h+ β1

)

=

(α0

1

hH(dh)|β0

dh

h

)+

(α0

1

hH(dh)|β1

)+

(H(α1)|β0

dh

h

)+ (H(α1|β1))

=α0

h(H(dh)|β1) +

β0h

(H(α1)|dh) + (H(α1)|β1)(H(β)|α

)=

=

(β0H(dh)

h+H(β1)|α0

dh

h+ α1

)

=

(β0H(dh)

h|α0

dh

h

)+α0

h(H(β1)|dh) +

β0h

(H(dh)|α1)) + (H(β1)|α1)

=α0

h(H(β1)|dh) +

β0h

(H(dh)|α1)) + (H(β1)|α1)(H(α)|β

)+(H(β)|α

)=

=α0

h((H(dh)|β1) + (H(β1)|dh)) +

β0h

((H(dh)|α1) + (H(α1)|dh))++(H(β1)|α1) + (β1|H(α1))

= 0 + 0 + 0 + 0

♦ù s♦tr♦♣ Gr(H).

é♠♦♥strt♦♥ ♦r♦r

st r q r♦t é♥t ♥s OX ② é♥t ♥ strtr èr

rst ♦♥ à érr ♥tté ♦ sr s st♦♥s rst♥ts MD.

t♣

♥ s ♦♥♥ u, v ∈ MD −OX t a ∈ OX ♦rs

u, v, aDD = u, 1vv, asD

=1

uvu, v, ass −

1

uv2u, vsv, as.

s♥st ♦♥ q

u, v, aDD+ =1

uvu, v, ass − 1

uv2u, vsv, as +

1

uvv, a, uss −

1

u2va, usv, us +

1

uva, u, vss −

1

uv2u, vsa, vs −

1

u2vu, vsa, us

t♣

♥ s ♦♥♥ v ∈ MD −OX t a, b ∈ OX . ♦rs

a, b, vDD = a, 1vb, vsD =

1

va, b, vss −

1

v2b, va, vs.

♠ê♠ ♥♦s ♦♥s

b, v, aDD = b, 1vv, asD

=1

vb, v, ass −

1

v2v, asb, vs

tv, a, bDD = v, a, bsD

=1

vv, a, bss

.

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

♥ ♥ ét ♦♥ q

a, b, vDD+ =1

va, b, vss −

1

v2b, va, vs+

+1

vb, v, ass −

1

v2v, asb, vs +

1

vv, a, bss

=1

va, b, vss +

1

vb, v, ass +

1

vv, a, bss

= 0

t♣

Pr♥♦♥s u, v, w ∈ MD −OX .

♦rs u, v, wDD = u, 1

vwv, wsD.

s1

vwv, ws ∈ OX ; r

1

vwv, ws = v, wD ∈ OX . ♥ rést q

u, v, wDD = u, 1

vwv, wsD.

=1

uvwu, v, wss −

1

uvw2v, wsu,ws −

1

uwv2v, wsu, vs.

♥ ♥ ét q

u, v, wDD+ =1

uvwu, v, wss −

1

uvw2v, wsu,ws −

1

uwv2v, wsu, vs

+1

uvwv, w, uss −

1

vwu2w, usv, us −

1

vuw2w, usv, ws

+1

uvww, u, vss −

1

wuv2u, vsw, vs −

1

wvu2u, vsw, us

=1

uvw(u, v, wss + v, w, uss + w, u, vss)

= 0.♥s −,−D stst ♥tté ♦ ♥ ♥ ét ♦♥ q st ♥ strtr

èr

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦♥t α, β ∈ ΩX(logD) ♦♥

Hα(Hβ) =α1β1h2

h, h,−+ α1

h2h, β1h,−+ α1β

j

hh, xj ,−+ α1

hh, βjxj ,−+

αiβ1h

xi, h,−+ αi

hxi, βjh,− − αiβj

h2xi, hh,−+ αiβjxi, xj ,−+

αixi, βjxj ,−.Hβ(Hα) =β1α1

h2h, h,−+ β1

h2h, α1h,−+ β1

hh, αixi,−+ β1α

i

hh, xi,−+

βjα1

hxj , h,−+ βj

h2xj , α1h,− −+

βjα1

h2xj , hh,−+ βjαixj , xi, +

βjxj , αixi,−.

t

♦♥Hα(Hβ)− Hβ(Hα) =

=α1

h2h, β1h,−+ αiβ1

hxi, h,−+ α1β

j

hh, xj,−+ α1

h2h, βjh,−+

β1h2

α1, hh,−+ β1hαi, hxi,−+ αi

hxi, βjh,−+ βj

hα1, xjh,−−

αiβæ

h2xi, hh,− − α1β

j

h2h, xjh,−+ αiβjxi, xj,−+ αixi, βjxj ,−+

βjαi, xjxi,−

Pr rs ♥♦s ♦♥sH([α, β]) =

=α1

h2h, β1h,−+ β1

h2α1, hh,−+ α1

hh, βjxi,−+ βj

hα1, xjh,−

+α1β

j

hh, xj,− − α1β

j

h2h, xjh,−+ αi

hxi, βjh,−+ β1

hαi, hxi,−+

αiβ1h

xi, h,− − αiβj

h2xi, hh, ,−+ αixi, βjxj ,−+ βjαi, xjxi,−+

αiβjxi, xj,−♥ ♥

H([α, β]) = [Hα, Hβ]

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

st qst♦♥ ♠♥r C[y]dx

x⊕ΩA ♥ strtr èr r strtr

P♦ss♦♥ s♥t

[dx, dy] := dx

t ΩA ♥ èr

②♥t st ①t ♦rt A♠♦s s♥t

0 // ΩA// ΩA ⊕ C[y]

dx

x// C[y]

dx

x// 0

st ♠ttr sr C[y]dx

x♥ strtr èr ♠♥èr à ♥ r ♥

①t♥s♦♥ s♥é r ♣rès ❬ss② t ❪

[γ1 + β1, γ2 + β2] = [γ1, γ2] + [β1, γ2]− [β2, γ1] + [β1, β2]

♦ù γi + βi ∈ ΩA ⊕ C[y]dx

x♣♦r i = 1, 2.

st ♥ strtr èr ♥s ΩA ⊕ C[y]dx

xà ♦♥t♦♥ q ΩA ♥ é

ΩA ⊕C[y]dx

x. st ♦♥ ♠♦♥trr q s r♦ts é♥s ♣r t s♦♥t

é①

P♦s♦♥s γ1 = γ01dx

x, β1 = β0

1dx+ β11dy t γ2 = γ02

dx

x, β2 = β0

2dx+ β12dy.

Pr ♥ rt ♦♥ ♦t♥t

[γ1, γ2] =

(γ01xx, γ02+

γ02xγ01 , x

)dx

x,

[β1, β2] =(β01x, β0

2+ β02β0

1 , x+ β12β0

1 , y+ β11y, β0

2+ (β01β

12 − β1

1β02))dx +(

β01x, β1

2+ β02β1

1 , x+ β11y, β1

2+ β12β1

1 , y)dy,

[β1, γ2] =γ02xβ0

1 , xdx+γ02xβ1

1 , xdy + (β01x, γ02+ β1

1y, γ02)dx

x,

[β2, γ1] =γ01xβ0

2 , xdx+γ01xβ1

2 , xdy + (β02x, γ01+ β1

2y, γ01)dx

x.

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

s étés q ♣réè♥t ♦♥ ét[γ1 + β1, γ2 + β2] = [γ1, γ2] + [β1, γ2]− [β2, γ1] + [β1, β2] =(γ01xx, γ02+

γ02xγ01 , x+ β0

1x, γ02+ β11y, γ02 − β0

2x, γ01 − β12y, γ01

)dx

x+

(γ02xβ0

1 , x −γ01xβ0

2 , x+ β01x, β0

2+ β02β0

1 , x+ β12β0

1 , y+ β11y, β0

2+ (β01β

12 − β1

1β02)

)dx+

(γ02xβ1

1 , x −γ01xβ1

2 , x+ β01x, β1

2+ β02β1

1 , x+ β11y, β1

2+ β12β1

1 , y)dy.

Pr rs γ1+β1 = (γ01 +xβ01)dx

x+β1

1dy, γ2+β2 = (γ02 +xβ02)dx

x+β1

2dy. ♥ ♣♣q♥t

r♦t é♥ ♣r ♦♥ ♦t♥t

[γ1 + β1, γ2 + β2] =

= [(γ01 + xβ01)dx

x, (γ02 + xβ0

2)dx

x] + [(γ01 + xβ0

1)dx

x, β1

2dy]

+ [β11dy, (γ

02 + xβ0

2)dx

x] + [β1

1dy, β12dy]

=(γ01 + xβ0

1)

xx, γ02 + xβ0

2dx

x+

(γ02 + xβ02)

xγ01 + xβ0

1 , xdx

x

+(γ01 + xβ0

1)

xx, γ12dy + β1

2γ01 + xβ01 , y

dx

x

+ β11y, γ02 + xβ0

2dx

x+γ02 + xβ0

2

xβ1

1 , xdy+ β1

1y, β12dy + β1

2β12 , ydy

= (γ01xx, γ02+ β0

1x, γ02+γ02xγ01 , x+ β0

2γ01 , x+ β12γ01 , y+ β1

1y, γ02)dx

x+ (β0

1x, β02+ γ01x, β0

2+ β02β0

1 , x+ γ02β01 , x+ β1

2β01 , y+ βy, β0

2+β12β

01 − β1

1β02)dx

+ (γ01xx, β1

2+ β01x, β1

2+γ02xβ1

1 , x+ β02β1

1 , x+ β11y, β1

2+ β12β1

1 , y)dy.

♥ ♥ été ré

é♠♦♥strt♦♥ ♠♠

♦♥t α, β t a ♦♠♠ ♥s ②♣♦tès ♠♠ ♥

[α, aβ] =

aα01

xx, β0

1dx

x+α01β

01

xx, adx

x+aβ0

1

xα0

1, xdx

x+α01a

xx;β1

1dy+

+α01β

11

xx, ady + aβ1

1α01, y

dx

x+ α1

1ay, β01dx

x+ α1

1β01y, a

dx

x+

+ aβ01α1

1, xdy + α11ay, β1

1dy + α11β

11y, ady + aβ1

1α11; ydy.

=

a(α01

xx, β0

1dx

x+β01

xα0

1, xdx

x+α01

xx;β1

1dy + β11α0

1, ydx

x

+ α11y, β0

1dx

x+ β0

1α11, xdy + α1

1y, β11dy + β1

1α11; ydy)

+ ((α01

xx, a+ α1

1y, a)β01

dx

x+ (

α01

xx, a+ α1

1y, a)β11dy)

=

a(α01

xx, β0

1dx

x+β01

xα0

1, xdx

x+α01

xx;β1

1dy + β11α0

1, ydx

x

+ α11y, β0

1dx

x+ β0

1α11, xdy + α1

1y, β11dy + β1

1α11; ydy)

+ (α01

xx, a+ α1

1y, a)β= H(α)(a).β + a[α, β].

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

♦♥t α = α01

dx

x+ α1

1dy t β = β01

dx

x+ β1

1dy ♥s ∈ ΩA(log I). ♥

H([α, β]) =1

x(α01

xx, β0

1+β01

xα0

1, x+ α11y, β0

1+ β11α0

1, y)x,−

(α01

xx, β1

1+β01

xα1

1, x+ α11y, β1

1+ β11α1

1, y)y,−

Pr rs ♦♥

H(α)H(β) =α01β

01

x2x, x,−+ α0

1

x2x, β0

1x,−+ α01β

11

xx, y,−+

α01

xx, β1

1y,−+ α11β

01

xy, x,−+ α1

1

xy, β0

1x,−+

−α11β

01

x2y, xx,−+ α1

1β11y, y,−+ α1

1y, β11y,−

H(β)H(α) =β01α

01

x2x, x,−+ β0

1

x2x, α0

1x,−+ β01α

11

xx, y,−+

β01

xx, α1

1y,−+ β11α

01

xy, x,−+ β1

1

xy, α0

1x,−+

−β11α

01

x2y, xx,−+ β1

1α11y, y,−+ β1

1y, α11y,−

H(α)H(β)− H(β)H(α) = H([α, β]) +

α11β

01

x(y, x,− − 1

xy, xx,− − x, y,−)

α11β

01

x(x, y,− − y, x,−+ 1

xy, xx,−).

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

r

y, x,− − 1

xy, xx,− − x, y,− =

= (y, x,−+ x, −, y+ x,−)= (y, x,−+ x, −, y+ −, y, x)= 0

t

x, y,− − y, x,−+ 1

xy, xx,− =

= x, y,−+ y, −, x − x,−= x, y,−+ y, −, x+ −, x, y= 0

♦ù réstt

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

st qst♦♥ ♠♦♥trr q qq s♦t α0, α1 t α2 ♥s ΩA(log I) ♦♥ 0 =

dρω (ω)(α0, α1, α2) = ρω(α0)ω(α1, α2)−ρω(α1)ω(α0, α2)+ρω(α2)ω(α0, α1)−ω([α0, α1], α2)+

ω([α0, α2], α1)−ω([α1, α2], α0.) st r sr s éé♠♥ts é♥értrs ΩA(log I).P♦r α0 =

du0u0

, α1 =du1u1

, α2 =du2u2

♦♥

dρω (ω)(α0, α1, α2) =1

u0u0,

1

u1u2u1, u2 −

1

u1u1,

1

u0u2u0, u2+

1

u2u2,

1

u0u1u0, u1

− 1

u2 1

u0u1u0, u1, u2+

1

u1 1

u0u2u0, u2, u1 −

1

u0 1

u1u2u1, u2, u0

=1

u0u1u2u0, u1, u2 −

1

u0u1u22u1, u2u0, u2

− 1

u0u2u21u1, u2u0, u1 −

1

u1u0u2u1, u0, u2+

1

u2u1u20u0, u2u1, u0

+1

u22u1u0u0, u2u1, u2+

1

u0u1u2u2, u0, u1 −

1

u2u1u20u0, u1u2, u0

− 1

u2u0u21u0, u1u2, u1+

1

u2u0u1u0, u1, u2+

1

u2u20u1u0, u1u0, u2

+1

u1u0u2u0, u2, u1 −

1

u1u20u2u0, u2u0, u1 −

1

u1u0u22u0, u2u2, u1

− 1

u0u1u2u1, u2, u0+

1

u0u21u2u1, u2u1, u0+

1

u0u1u22u1, u2u2, u2.

♥ ♣♣q♥t ① ♦s ♥tté ♦ ♦♥ ♦t♥t dρω (ω)(α0, α1, α2) = 0. ♠ê♠

♣♦r α0 =du0u0

, α2 =du1u1

, α2 = du2

dρω (ω)(α0, α1, α2) =1

u0u0,

1

u1u1, u2 −

1

u1u1,

1

u0u0, u2+ u2,

1

u0u1u0, u1

=1

u0u1u0, u1, u2 −

1

u0u21u1, u2u0, u1 −

1

u1u0u1, u0, u2+

1

u1u20u0, u2u1, u0

+1

u0u1u2, u0, u1 −

1

u0u21u0, u1u2, u1 −

1

u20u1u0, u1u2, u0 −

1

u0u1u0, u1, u2

+1

u0u21u0, u1u1, u2+

1

u20u1u0, u1u0, u2+

1

u1u0u0, u2, u1 −

1

u1u20u0, u2u0, u1

− 1

u0u1u1, u2, u0+

1

u0u21u1, u2u1, u0 = 0.

♠ê♠ ♦♥ ♠♦♥tr dρω (ω)(α0, α1, α2) = 0 ♣♦r α0 =du0u0

, α1 = du1, α2 = du2 ♥s q

♣♦r α0 = du0, α1 = du1, α2 = du2.

é♠♦♥strt♦♥ Pr♦♣♦st♦♥

P♦r ωi = aiduiui

+ bidvi ωj = ajdujuj

+ bjdvj t f ∈ A ♦♥

[ωi, fωj ] = ρω(ωi)(a)ωj + f [ωi, ωj ]

♥ t

♥♥① P♦♥ts ét qqs é♠♦♥strt♦♥s

[ωi, fωj ] = [aiduiui, faj

dujuj

] + [aiduiui, fbjdvj ] + [bidvi, faj

dujuj

]

=aiui

ui, faj+faiuj

ai, ujduiui

+ faiajd(1

uiujui, uj) +

aiui

ui, fbjdvj+

fbjai, vjduiui

+ faid(1

uiui, vj) + bivi, faj

dujuj

+fajuj

bi, ujdvi+

fbiajd(1

ujvi, uj) + bivi, fbjdvj + fbjbi, vjdvi + fbibjd(vi, vj)

=faiui

ui, ajdujuj

+aiajui

ui, fdujuj

+fajuj

ai, ujduiui

+

faiajd(1

uiujui, uj) +

faiui

ui, bjdvj +aibjui

ui, fdvj+

fbjai, vjduiui

+ faibjd(1

uiui, vj) + bifvi, aj

dujuj

+

fbivi, bjdvj + bibjvi, fdvj + fbjbi, vjdvi + fbibjd(vi, vj)= f(

aiui

ui, ajdujuj

+ajuj

ai, ujduiui

+ aiajd(1

uiujui, uj)

aiui

ui, bjdvj + bjai, vjduiui

+ aibjd(1

uiui, vj)

bivi, bjdvj +aiui

bi, ujdvi + biajd(1

ujvi, uj)+

bivi, bjdvj + bjbi, vjdvi + bibjd(vi, vj))+= f(

aiui

ui, ajdujuj

+ajuj

ai, ujduiui

+ aiajd(1

uiuj)

aiui

ui, bjdvj + bjai, vjduiui

+ aibjd(1

uiui, vj)

bivi, bjdvj +aiui

bi, ujdvi + biajd(1

ujvi, uj)+

bivi, bjdvj + bjbi, vjdvi + bibjd(vi, vj))+[(aiui

ui,−+ bjui

ui,−dvj + biajvi,−+ bjvi,−dvj)(f)

](aj

dujuj

+ bjdvj)

= f [ωi, ωj ] + (ρω(ωi)(f))ωj .

♦ù réstt

♥♥①

s ♣♦♥ts s qqs

s

♥tr♦t♦♥

t ♥♥① ♦r♥t qq éts sr s ♣♦♥ts s rt♥s ♥ ér

♥♦t♠♠♥t ♥♦t♦♥ étté ér♥t ♦♥strt ♣tr

s strtr f, g = xyzdf ∧ dg ∧ dp

dx ∧ dy ∧ dz

♥s s ♥♦s ♥♦s ♦♥♥♦♥s ♥ ♣♦②♥ô♠ ♥♦♥ ♦♥st♥t p ♥s A = C[x, y, z] râ

q ♥♦s é♥ss♦♥s r♦t P♦ss♦♥ ♦rt♠q s♥t

f, g = hdf ∧ dg ∧ dpdx ∧ dy ∧ dz

P♦r ér s ♥♦tt♦♥s ♥♦s ♦♥sér♦♥s s s♦♠♦r♣s♠s s♥ts

Ω1A(logD)

ϕ1−→ A3 ∼= A×A×Af1dx

x+ f2

dy

y+ f3

dz

z7→ (f1, f2, f3)

Ω2A(logD)

ϕ2−→ A3 ∼= A×A×Af1dy

y∧ dz

z+ f2

dz

z∧ dx

x+ f3

dx

x∧ dy

y7→ (f1, f2, f3)

Ω3A(logD)

ϕ2−→ Afdx

x∧ dy

y∧ dz

z7→ f

∧1DerA(logD)ψ1−→ A3 ∼= A×A×A

f1x∂x+ f2y∂y + f3z∂z 7→ (f1, f2, f3)

∧2DerA(logD)ψ2−→ A3 ∼= A×A×A

f1y∂y ∧ z∂z + f2z∂z ∧ x∂x+ f3x∂x ∧ y∂y 7→ (f1, f2, f3)

∧3DerA(logD)ψ3−→ A

fx∂x ∧ y∂y ∧ z∂z 7→ f

râ à s s♦♠♦r♣s♠s s ♦♣értrs é♥s ♣r éqt♦♥ ♥♥♥t

∂0f = ∂xh(∂yf∂zp− ∂zf∂yp)x∂x + ∂yh(∂zf∂xp− ∂xf∂zp)y∂y+

∂zh(∂xf∂yp− ∂yf∂xp)z∂z

♥♥① s ♣♦♥ts s qqs s

P♦r t♦t f ∈ A,

∂1 ~f =

∂yh(∂zf3∂xp− ∂xf3∂zp)− ∂zh(∂xf2∂yp− ∂yf2∂xp)

−f1x2∂2xxp− f2xy∂2xyp− f3xz∂

2xzp− f1x∂xp

∂zh(∂xf1∂yp− ∂yf1∂xp)− ∂xh(∂yf3∂zp− ∂zf3∂yp)

−f1xy∂2xyp− f2y2∂2yyp− f3yz∂

2yzp− f2y∂yp

∂xh(∂yf2∂zp− ∂zf2∂yp)− ∂yh(∂zf1∂xp− ∂xf1∂zp)

−f1xz∂2xzp− f2yz∂2yzp− f3z

2∂2zzp− f3z∂zp

P♦r t♦t ~f ∈ A3 t ♥ ♥

∂2 ~f = ∂xh(∂yf1∂zp− ∂zf1∂yp) + ∂yh(∂zf2∂xp− ∂xf2∂zp)+

∂zh(∂xf3∂yp− ∂yf3∂xp)

P♦r t♦t ~f ∈ A3. P♦s♦♥s Pi : A3 → A ♣r♦t♦♥ sr iè♠ ♦♠♣♦s♥t

♦♥tr♦♥s q ∂1 ∂0 = 0

♦t f ∈ A.

∂0(f) =

f1 = z∂zpy∂yf − y∂ypz∂zf

f2 = x∂xpz∂zf − z∂zpx∂xf

f3 = y∂ypx∂xf − x∂xpy∂yf

♣rès ♣r♠èr ♦♠♣♦s♥t p1(∂1(∂0(f))) st ♦♥♥é ♣r

P1(∂1(∂0(f))) = ∂yh(∂zf3∂xp− ∂xf3∂zp)− ∂zh(∂xf2∂yp− ∂yf2∂xp)

−f1x2∂2xxp− f2xy∂2xyp− f3xz∂

2xzp− f1x∂xp

♥ sstt♥t f1, f2, f3 ♣r rs ①♣rss♦♥s sss ♦♥ ♦t♥tP1(∂1(∂0(f))) = x2zy∂xp∂xf∂

2yzp+ x2yz∂xp∂yp∂

2xzf − x2yz∂xp∂yf∂

2zxp− x2yz(∂xp)

2∂2zyf−x2yz∂zp∂xf∂

2xyp− x2yz∂zp∂yp∂

2xxf − xzy∂zp∂yp∂xf + x2yz∂zp∂yf∂

2xxp+ x2yz∂zp∂xp∂

2xyf+

xyz∂zp∂xp∂yf − x2yz∂yp∂zf∂2xxp− x2yz∂yp∂xp∂

2xzf − xyz∂yp∂xp∂zf + x2yz∂yp∂xf∂

2xzp+

x2yz∂yp∂zp∂2xxf + xyz∂yp∂zp∂xf + x2yz∂xp∂zf∂

2xyp+ x2yz(∂xp)

2∂2yzf − x2yz∂xp∂xf∂2yz−

x2yz∂xp∂zp∂2yxf − x2z∂zp∂yf∂

2xxp+ x2yz∂yp∂zf∂

2xxp− x2yz∂xp∂zf∂

2xyp+ x2yz∂zp∂xf∂

2xyp

−x2yz∂yp∂xf∂2xzp+ x2yz∂xp∂yf∂2xzp− xyz∂zp∂yf∂xp+ xyz∂yp∂zf∂xp

= 0 ç♦♥ ♥♦ ♦♥ ♠♦♥tr q s trs ♦♠♣♦s♥ts s♦♥t t♦ts ♥s

♦♥tr♦♥s q ∂2 ∂1 = 0

P♦r ér s s t tr tr ♦♥ ♣♦s ♣♦r t♦t ~f =

(f1, f2, f3); (F1, F2, F3) = ~F = ∂1(~f).

♦rs ∂1 ~f =

F1 = ∂yh(∂zf3∂xp− ∂xf3∂zp)− ∂zh(∂xf2∂yp− ∂yf2∂xp)

−f1x2∂2xxp− f2xy∂2xyp− f3xz∂

2xzp− f1x∂xp

F2 = ∂zh(∂xf1∂yp− ∂yf1∂xp)− ∂xh(∂yf3∂zp− ∂zf3∂yp)

−f1xy∂2xyp− f2y2∂2yyp− f3yz∂

2yzp− f2y∂yp

F3 = ∂xh(∂yf2∂zp− ∂zf2∂yp)− ∂yh(∂zf1∂xp− ∂xf1∂zp)

−f1xz∂2xzp− f2yz∂2yzp− f3z

2∂2zzp− f3z∂zpP♦s♦♥s ♥♥

δ1 = −∼−→A 1; δ2 = −

∼−→A 2; δ3 = −

∼−→A 3

♣rès ♦♥ r ∂2(~F ) = δ1(F1) + δ2(F2) + δ3(F3) ♦rs

s strtr P♦ss♦♥ x, y = x.

F1 = δ2(f3)− δ3(f2)− f1x2∂2xxp− f2xy∂

2xyp− f3xz∂

2xzp− f1x∂xp

F2 = δ3(f1)− δ1(f3)− f1xy∂2yxp− f2y

2∂2yyp− f3yz∂2yzp− f2y∂yp

F3 = δ1(f2)− δ2(f1)− f1xz∂2zxp− f2yz∂

2zyp− f3z

2∂2zzp− f3z∂zpt ♦♥

∂2(~F ) =

δ1 δ2(f3)− δ1 δ3(f2)− δ1(f1x2∂2xxp)− δ1(f2xy∂

2xyp)− δ1(f3xz∂

2xzp)− δ1(f1x∂xp)+

δ2 δ3(f1)− δ2 δ1(f3)− δ2(f1xy∂2yxp)− δ2(f2y

2∂2yyp)− δ2(f3yz∂2yzp)− δ2(f2y∂yp)+

δ3 δ1(f2)− δ3 δ2(f1)− δ3(f1xz∂2zxp)− δ3(f2yz∂

2zyp)− δ3(f3z

2∂2zzp)− δ3(f3z∂zp)+ st ♦♥ r q tr♠ tt ①♣rss♦♥ t érr q♦♥ ♦t♥t

t♠♥t ③ér♦ ♦t t ♦♥ ♦t♥t

δ1 δ2(f3) = xyz2(∂zp∂2xyp∂z + ∂zp∂xp∂

2yz − ∂yp∂

2xzp∂z − ∂yp∂xp∂

2zz − ∂zp∂

2yzp∂x

−(∂zp)2∂2yx + ∂yp∂

2zzp∂x + ∂yp∂zp∂

2xz)f3 + xyz(∂yp∂zp∂x − ∂yp∂xp∂z)f3

δ2 δ1(f3) = xyz2(∂xp∂2zzp∂y + ∂zp∂

2xyp∂z − ∂xp∂yp∂2zz − ∂zp∂

2xzp∂y − ∂xp∂

2yzp∂z

−(∂zp)2∂2yx + ∂zp∂yp∂

2zx − ∂zp∂xp∂

2yz)f3 + xyz(∂xp∂zp∂y − ∂xp∂yp∂z)f3

δ1 δ3(f2) = xy2z(∂zp∂2yyp∂x + ∂zp∂yp∂

2yx + ∂yp∂

2xzp∂y + ∂yp∂xp∂

2yz − ∂y∂

2yzp∂x

−(∂yp)2∂2xz − ∂zp∂

2xyp∂y − ∂zp∂xp∂

2yy)f2 + xyz(∂zp∂yp∂x − ∂zp∂xp∂y)f2

δ3 δ1(f2) = xy2z(∂yp∂2xzp∂y + ∂yp∂zp∂

2yx + ∂xp∂

2yyp∂z + ∂xp∂yp∂

2yz − ∂x∂

2yzp∂y

−(∂yp)2∂2xz − ∂yp∂

2xyp∂z − ∂xp∂zp∂

2yy)f2 + xyz(∂xp∂yp∂z − ∂xp∂zp∂y)f2

δ2 δ3(f1) = x2yz(∂xp∂2zyp∂x + ∂xp∂yp∂

2xz + ∂zp∂xp∂

2xy + ∂zp∂

2xxp∂y − ∂xp∂

2xzp∂y

−(∂xp)2∂2yz − ∂zp∂yp∂

2xx − ∂zp∂

2yxp∂x)f1 + xyz(∂zp∂xp∂yp− ∂zp∂yp∂x)f1

δ3 δ2(f1) = x2yz(∂yp∂2xxp∂z + ∂yp∂xp∂

2xz + ∂xp∂zp∂

2xy + ∂xp∂

2yzp∂x − ∂xp∂

2xyp∂z

−(∂xp)2∂2yz − ∂yp∂zp∂

2xx − ∂yp∂

2zxp∂x)f1 + xyz(∂yp∂xp∂zp− ∂yp∂zp∂x)f1

−δ1(f1x∂x(x∂xp)) = −xyz∂zp∂yf1∂xp− x2yz∂yf1∂2xxp+ xyz∂yp∂zf1∂xp+

x2yz∂yp∂zf1∂2xxp− f1xyz∂zp∂

2xyp− f1x

2yz∂wp∂3xxyp+ f1xyz∂yp∂

2xzp+

f1x2yz∂yp∂

3xxzp

−δ1(f2xy∂2xyp) = −xy2z∂zp∂2xyf2 + xy2z∂yp∂2xyp∂zf2 − f2xyz∂zp∂

2xyp

−f2xy2z∂zp∂3xyyp+ f2xy2z∂yp∂

2xyzp

−δ1(f3xz∂2xzp) = −xz2y∂zp∂yf3∂2xzf2 + xz2y∂yp∂2xzp∂zf3 − f3xyz

2∂zp∂3xyzp

−f3xz2y∂yp∂3xzzp+ f3xyz∂yp∂2xzp

−δ2(f2y∂y(y∂yp)) = −xyz2∂xp∂yp∂zf2 + xyz∂zp∂yp∂xf2 − xy2z∂xp∂zf2∂2yyp

+xy2z∂zp∂xf2∂2yyp− f2xyz∂xp∂

2yzp+ f2xyz∂zp∂

2xyp− f2xy

2z∂xp∂yyzp+ f2xy2z∂zp∂

3xyyp

−δ2(f1xy∂2xyp) = −yx2z∂xp∂zf1∂2xyp+ yx2z∂zp∂2xyp∂xf1 − f1x

2yz∂xp∂3xyzp

+f1x2yz∂zp∂

3xxyp+ f1xyz∂zp∂

2xyp

−δ2(f3zy∂2zyp) = −xz2y∂xp∂2zyf3 + xz2y∂zp∂2zyp∂xf3 − f3xyz

2∂xp∂3yzzp

−f3xyz∂xp∂2zyp+ f3xz2y∂zp∂

3xyzp

−δ3(f3z∂z(z∂zp)) = −xyz∂yp∂zp∂xf3 − xyz2∂yp∂xf3∂2zzp+ xyz∂xp∂yf3∂zp

+xyz2∂xp∂yf3∂2zzp− f3xyz∂yp∂

2xzp− f3xyz

2∂yp∂2xzzp+ f3xyz∂xp∂

2yzp+ f3xyz

2∂xp∂3yzzp

−δ3(f1xz∂2xzp) = −x2yz∂yp∂xf1∂2xyp+ x2yz∂xp∂2xzp∂yf1 − f1x

2yz∂yp∂3xyxp

−f1xyz∂yp∂2xzp+ f1x2yz∂xp∂

3xyzp

−δ3(f2yz∂2yzp) = −y2xz∂yp∂xf2∂2zyp+ xy2z∂xp∂2yzp∂yf2 − f2xy

2z∂yp∂3xyzp

+f2xy2z∂xp∂

3yyzp+ f2xyz∂xp∂

2yzp

s strtr P♦ss♦♥ x, y = x.

♥s t ♥♥① ♥♦s ♥♦s ♣r♦♣♦s♦♥s érr sr ①♠♣ x, y = x tté

strtr èr ♥rt sr ΩA(log xA).

s éé♠♥ts ♠♦ ΩA(log xA) s♦♥t s♦s ♦r♠ αdx

x+ βdy ♦ù α, β ∈ A.

♥♥① s ♣♦♥ts s qqs s

♦♥t α1 = α01

dx

x+ α1

1dy, α2 = α02

dx

x+ α1

2dy, α3 = α03

dx

x+ α1

3dy tr♦s éé♠♥ts

ΩA(log xA).

♦rs

[α1, α2] = [α01

dx

x, α0

2

dx

x] + [α0

1

dx

x, α1

2dy] + [α11dy, α

02

dx

x] + [α1

1dy, α12dy].

r [α01

dx

x, α0

2

dx

x] = (α0

1∂yα02 − α0

2∂yα01)dx

x, [α0

1

dx

x, α1

2dy] = xα12∂xα

01

dx

x+ α0

1∂yα12dy,

[α11dy, α

02

dx

x] = −xα1

1∂xα02

dx

x− α0

2∂yα11dy, [α

11dy, α

12dy] = (xα1

2∂xα11 − xα1

1∂xα12)dy

s♥st q

[α1, α2] = (α01∂yα

02 − α0

2∂yα01 + xα1

2∂xα01 − xα1

1∂xα02)dx

x+ (α0

1∂yα12 − α0

2∂yα11 + xα1

2∂xα11 −

xα11∂xα

12)dy

P♦s♦♥s α = α01∂yα

02 −α0

2∂yα01 + xα1

2∂xα01 − xα1

1∂xα02 t β = α0

1∂yα12 −α0

2∂yα11 + xα1

2∂xα11 −

xα11∂xα

12.

♥ ♦rs

[[α1, α2], α3] = [αdx

x+ βdy, α0

3

dx

x+ α1

3dy]

= [αdx

x, α0

3

dx

x] + [α

dx

x, α1

3dy] + [βdy, α03

dx

x] + [βdy, α1

3dy]

r

[αdx

x, α0

3

dx

x] = (α∂yα

03 − α0

3∂yα)dx

x, [α

dx

x, α1

3dy] = xα13∂xα

dx

x+ α∂yα

13dy,

[βdy, α03

dx

x] = −xβ∂xα0

3

dx

x− α0

3∂yβdy, [βdy, α13dy] = (xα1

3∂xβ − xβ∂xα13)dy.

♦♥

[[α1, α2], α3] = (α∂yα03 − α0

3∂yα + xα13∂xα + xβ∂xα

03)dx

x+ (α∂yα

13 − α0

3∂yβ + xα13∂xβ −

xβ∂xα13)dy.

♥ ♦♥sèr s ♣♣t♦♥s Pi : ΩA(log xA) → A é♥s ♣r

P1(adx

x+ bdy) = a t P2(a

dx

x+ bdy)) = b t ♦♥ ♣♦s

A123 := P1([[α1, α2], α3]), A231 := P1([[α2, α3], α1]) t A312 := P1([[α3, α1], α2]).

Pr rs s ♥♦s ♣♦s♦♥s B123 := P2([[α1, α2], α3]), B231 := P2([[α2, α3], α1]) t

B312 := P2([[α3, α1], α2]),

♦rs

A123 = α∂yα03 −α0

3∂yα+ xα13∂xα+ xβ∂xα

03 t B123 = α∂yα

13 −α0

3∂yβ + xα13∂xβ − xβ∂xα

13.

Pr rs∂y(α) = ∂y(α

01∂yα

02 − α0

2∂yα01 + xα1

2∂xα01 − xα1

1∂xα02)

= ∂yα01∂yα

02 + α0

1∂2yyα

02 − ∂yα

02∂yα

01 − α0

2∂2yyα

01 + x∂yα

12∂xα

01+

xα12∂

2yxα

01 − x∂yα

11∂xα

02 − xα1

1∂2xyα

02

∂xα = ∂x(α01∂yα

02 − α0

2∂yα01 + xα1

2∂xα01 − xα1

1∂xα02)

= ∂xα01∂yα

02 + α0

1∂2xyα

02 − ∂xα

02∂yα

01 − α0

2∂2xyα

01 + α1

2∂xα01 + x∂xα

12∂xα

01 + xα1

2∂2xxα

01

−α11∂xα

02 − x∂xα

11∂xα

02 − xα1

1∂2xxα

02

s strtr P♦ss♦♥ x, y = x.

♥ ♦t♥t ♦♥

A123 = α01∂yα

02∂yα

03 − α0

2∂yα01∂yα

03 + xα1

2∂xα01∂yα

03 − xα1

1∂xα02∂yα

03 − α0

3∂yα01∂yα

02

−α03α

01∂

2yyα

02 + α0

3∂yα02∂yα

01 + α0

3α02α

2yyα

01 − xα0

3∂yα12∂xα

01 − xα0

3α12∂

2yxα

01 + xα0

3∂yα11∂xα

02+

xα03α

11∂

2xyα

02 ++xα1

3∂xα01∂yα

02 + xα1

3α01∂

2xyα

02 − xα1

3∂xα02∂yα

01 − xα1

3α02∂

2xyα

01 + xα1

3α12∂xα

01+

x2α13∂xα

12∂xα

01 + x2α1

3α12∂

2xxα

01 − xα1

3α11∂xα

02 − x2α1

3∂xα11∂xα

02 − x2α1

3α11∂

2xxα

02 − xα0

1∂yα12∂xα

03+

xα02∂yα

11∂xα

03 + x2α1

1∂xα12∂xα

03 − x2α1

2∂xα11∂xα

03

A231 = α02∂yα

03∂yα

01 − α0

3∂yα02∂yα

01 + xα1

3∂xα02∂yα

01 − xα1

2∂xα03∂yα

01 − α0

1∂yα02∂yα

03

−α01α

02∂

2yyα

03 + α0

1∂yα03∂yα

02 + α0

1α03∂

2yyα

02 − xα0

1∂yα13∂xα

02 − xα0

1α13∂

2yxα

02 + xα0

1∂yα12∂xα

03+

xα01α

12∂

2xyα

03 ++xα1

1∂xα02∂yα

03 + xα1

1α02∂

2xyα

03 − xα1

1∂xα03∂yα

02 − xα1

1α03∂

2xyα

02 + xα1

1α13∂xα

02+

x2α11∂xα

13∂xα

02 + x2α1

1α13∂

2xxα

02 − xα1

1α12∂xα

03 − x2α1

1∂xα12∂xα

03 − x2α1

1α12∂

2xxα

03 − xα0

2∂yα13∂xα

01+

xα03∂yα

12∂xα

01 + x2α1

2∂xα13∂xα

01 − x2α1

3∂xα12∂xα

01

A312 = α03∂yα

01∂yα

02 − α0

1∂yα03∂yα

02 + xα1

1∂xα03∂yα

02 − xα1

3∂xα01∂yα

02 − α0

2∂yα03∂yα

01

−α02α

03∂

2yyα

01 + α0

2∂yα01∂yα

03 + α0

2α01∂

2yyα

03 − xα0

2∂yα11∂xα

03 − xα0

2α11∂

2yxα

03 + xα0

2∂yα13∂xα

01+

xα02α

13∂

2xyα

01 ++xα1

2∂xα03∂yα

01 + xα1

2α03∂

2xyα

01 − xα1

2∂xα01∂yα

03 − xα1

2α01∂

2xyα

03 + xα1

2α11∂xα

03+

x2α12∂xα

11∂xα

03 + x2α1

2α11∂

2xxα

03 − xα1

2α13∂xα

01 − x2α1

2∂xα13∂xα

01 − x2α1

2α13∂

2xxα

01 − xα0

3∂yα11∂xα

02+

xα01∂yα

13∂xα

02 + x2α1

3∂xα11∂xα

02 − x2α1

1∂xα13∂xα

02

♦ù A123 +A231 +A312 = 0

P♦r ♠♦♥trr qB123+B231+B312 = 0 ♦♥ ♣t ♣r♦ér ♦♠♠ sss ♥ r♠♣ç♥t

α t β ♣r rs ①♣rss♦♥s rs♣ts ♦s ♦♥s ♣r♦ér tr♠♥t é st tsr

♥tté ♦ strtr P♦ss♦♥ s♦♥t

♠rq♦♥s q

[[α1, α2], α3] = [[α01

dx

x, α0

2

dx

x], α0

3

dx

x] + [[α0

1

dx

x, α0

2

dx

x], α1

3dy] + [[α01

dx

x, α1

2dy], α03

dx

x] +

[[α01

dx

x, α1

2dy], α13dy] + [[α1

1dy, α02

dx

x], α0

3

dx

x] + [[α1

1dy, α02

dx

x], α1

3dy]

[[α11dy, α

12dy], α

03

dx

x] + [[α1

1dy, α12dy], α

13dy]

♥ ♥st ♠♠ s♥t

♠♠ s ♥♦tt♦♥s sss ♦♥

[[α01

dx

x, α0

2

dx

x]+ = 0

[[α11dy, α

12dy], α

13dy]+ = 0

Pr

P♦r q st ♣r♠èr été ♦♥

♥♥① s ♣♦♥ts s qqs s

[[α01

dx

x, α0

2

dx

x]+ =

= ( 1x(α0

1

xx, α0

2+α0

2

xα0

1, x)x, α03+

α0

3

xα

0

1

xx, α0

2+α0

2

xα0

1, x, x)dx

x= ( 1

x(α0

1

xx, α0

2x, α03+

α0

2

xα0

1, xx, α03) +

α0

3

xx, α0

2α0

1

x, x+ α0

3

x

α0

1

xx, α0

2, x+α0

3

xα0

1, xα0

2

x, x+ α0

3

x

α0

2

xα0

1, x, x)dx

x+

= ( 1x(α0

1

xx, α0

2x, α03+

α0

2

xα0

1, xx, α03) +

α0

3

x2 x, α02α0

1, x −α0

3

x

α0

1

x2 x, α01x, x+

α0

3

x

α0

1

xx, α0

2, x+α0

3

x2 α01, xα0

2, x −α0

3

x

α0

2

x2 α01, xx, x+

α0

2

x

α0

3

xα0

1, x, x)dx

x+

= ( 1x(α0

1

xx, α0

2x, α03+

α0

2

xα0

1, xx, α03) +

α0

3

x

α0

1

xx, α0

2, x+α0

2

x

α0

3

xα0

1, x, x)dx

x+

= (α0

1

x2 x, α02x, α0

3+α0

2

x2 α01, xx, α0

3+α0

3

x

α0

1

xx, α0

2, x+α0

2

x

α0

3

xα0

1, x, xα0

2

x2 x, α03x, α0

1+α0

3

x2 α02, xx, α0

1+α0

1

x

α0

2

xx, α0

3, x+α0

3

x

α0

1

xα0

2, x, xα0

3

x2 x, α01x, α0

2+α0

1

x2 α03, xx, α0

2+α0

2

x

α0

3

xx, α0

1, x+α0

1

x

α0

2

xα0

3, x, x)dx

x= 0

P♦r q st ♦♥ [[α1

1dy, α12dy], α

13dy]+ =

= [(α11y, α1

2+ α12α1

1, y)dy, α13dy]+

= (α11y, α1

2+ α12α1

1, y)y, α13+ α1

3α11y, α1

2+ α12α1

1, y, y)dy+

= (α11y, α1

2y, α13+ α1

2α11, yy, α1

3+ α13α

11y, α1

2, y+ α13α

12α1

1, y, yα13y, α1

2α11, y+ α1

3α11, yα1

2, y +

α12y, α1

3y, α11+ α1

3α12, yy, α1

1+ α11α

12y, α1

3, y+ α11α

13α1

2, y, yα11y, α1

3α12, y+ α1

1α12, yα1

3, y +

α13y, α1

1y, α12+ α1

1α13, yy, α1

2+ α12α

13y, α1

1, y+ α12α

11α1

3, y, yα12y, α1

1α13, y+ α1

2α13, yα1

1, y)dy= 0

s♥st q s ♦♥ts dy rst♥t ♣r♦♥♥♥t

[[α01

dx

x, α0

2

dx

x], α1

3dy]+[[α01

dx

x, α1

2dy], α03

dx

x]+[[α0

1

dx

x, α1

2dy], α13dy]+[[α1

1dy, α02

dx

x], α0

3

dx

x]+

[[α11dy, α

02

dx

x], α1

3dy] + [[α11dy, α

12dy], α

03

dx

x]

P♦r tr♠♥r st ♠♦♥trr q s r♥èrs s♦♥t ♥s P♦r ♣r♦♦♥s

♠♠ s♥t

♠♠ ♦t 〈−,−〉 r♦t té DerA(log xA) = Ω∗A(log xA). ♦rs

〈[[α01

dx

x, α0

2

dx

x], α1

3dy] + [[α01

dx

x, α1

2dy], α03

dx

x] + [[α1

1dy, α02

dx

x], α0

3

dx

x], ∂y〉+ = 0

〈[[α01

dx

x, α1

2dy], α13dy] + [[α1

1dy, α02

dx

x], α1

3dy] + [[α11dy, α

12dy], α

03

dx

x], ∂y〉+ = 0

Pr

P♦r q st ♥♦s ♦♥s

〈[[α01

dx

x, α0

2

dx

x], α1

3dy] + [[α01

dx

x, α1

2dy], α03

dx

x] + [[α1

1dy, α02

dx

x], α0

3

dx

x], ∂y〉+ =

=α0

1

x2 x, α02x, α1

3+α0

2

x2 α01, xx, α1

3+α0

3

x

α0

1

xx, α1

2, x+α0

3

x2 x, α12α0

1, x+α0

3

x

α0

2

xα1

1, x, x+α0

3

x2 α11, xα0

2, x +α0

2

x2 x, α03x, α1

1+α0

3

x2 α02, xx, α1

1+α0

1

x

α0

2

xx, α1

3, x+α0

1

x2 x, α13α0

2, x+α0

1

x

α0

3

xα1

2, x, x+α0

1

x2 α12, xα0

3, x +α0

3

x2 x, α01x, α1

2+α0

1

x2 α03, xx, α1

2+α0

2

x

α0

3

xx, α1

1, x+α0

2

x2 x, α11α0

3, x+α0

2

x

α0

1

xα1

3, x, x+α0

2

x2 α13, xα0

1, x +

s strtr P♦ss♦♥ x, y = x.

♥t à ♦♥

〈[[α01

dx

x, α1

2dy], α13dy] + [[α1

1dy, α02

dx

x], α1

3dy] + [[α11dy, α

12dy], α

03

dx

x], ∂y〉+

=α0

1

xx, α1

2y, α13+

α1

2

xα0

1, yx, α13+

α1

3α0

1

xx, α1

2, y+α1

3

xx, α1

2α01, y+

−α1

3α0

1

x2 x, α12x, y+

α1

1

xy, α0

2x, α13+

α0

2

xα1

1, xy, α13+

α1

3α0

2

xα1

1, x, y+α1

3

xα1

1, xα02, y −

α1

3α0

2

x2 α11, xx, y+

α0

3α1

1

xy, α1

2, x+α0

3

xy, α1

2α11, x+

α0

3α1

2

xα1

1, y, x+α0

3

xα1

1, yα12, x

α0

2

xx, α1

3y, α11+

α1

3

xα0

2, yx, α11+

α1

1α0

2

xx, α1

3, y+α1

1

xx, α1

3α02, y+

−α1

1α0

2

x2 x, α13x, y+

α1

2

xy, α0

3x, α11+

α0

3

xα1

2, xy, α11+

α1

1α0

3

xα1

2, x, y+α1

1

xα1

2, xα03, y −

α1

1α0

3

x2 α12, xx, y+

α0

1α1

2

xy, α1

3, x+α0

1

xy, α1

3α12, x+

α0

1α1

3

xα1

2, y, x+α0

1

xα1

2, yα13, x

α0

3

xx, α1

1y, α12+

α1

1

xα0

3, yx, α12+

α1

2α0

3

xx, α1

1, y+α1

2

xx, α1

1α03, y+

−α1

2α0

3

x2 x, α11x, y+

α1

3

xy, α0

1x, α12+

α0

1

xα1

3, xy, α12+

α1

2α0

1

xα1

3, x, y+α1

2

xα1

3, xα01, y −

α1

2α0

1

x2 α13, xx, y+

α0

2α1

3

xy, α1

1, x+α0

2

xy, α1

1α13, x+

α0

2α1

1

xα1

3, y, x+α0

2

xα1

3, yα11, x

= 0

♠♥èr ♥♦ ♦♥ ♣r♦

♦r♣

❬ss② t ❪ ♠tr ss② Ptr ❲ ♦r t ❲♦♥ ♣♣rt ①

t♥s♦♥ ♦ rs r♥ r♦♥r ♥sttt t t♠ts P②s ♦t③

♠♥♥ss té ♥ ♣

❬t② t♥ ❪ r♥s t② t t♥ ♦♠tr② ♥ ②

♥♠s ♦ ♠♥t ♠♦♥♦♣♦s Pr♥t♦♥ ❯♥rst② Prss P♦rtr

trs té ♥ ♣

❬r ❪ P r ♣r♥♣s ♦ q♥t♠ ♠♥s ①♦r ❯♥rst② Prss

té ♥ ♣s t

❬♦♥s♦♥ ❪ ♠♦♥ r♥ ♦♥s♦♥ ♠s qt♦♥s ♥ t sst♦♥ ♦

♠♦♥♦♣♦s ♦♠♠♥ t P②s ♦ ♣s té ♥ ♣

❬♦♥♦ ❪ ♦s♣ ♦♥♦ ♦rt♠ P♦ss♦♥ ♦♦♠♦♦② ①♠♣ ♦ t♦♥

♥ ♣♣t♦♥ t♦ ♣rq♥t③t♦♥ ❳ ♣t ♥r té

♥ ♣

❬♦t♦ ❪ ②s ♦t♦ ♦③♥s②❲tt♥ ♥r♥ts ♦ ♦ ②♠♣t ♥♦s ♦♥

t♠♣♦rr② t♠ts ♦ ♣s té ♥ ♣s t

❬♦ss t ❪ ♦ss rtr♠ ♦st♥t t ① ♦s♥r r♥t

♦r♠s ♥ r ♥ rs r♥s ♠r t ♦ ♦ ♥♦ ♣s

rs té ♥ ♣

❬s♠♥♥ ❪ ♦♥♥s s♠♥♥ P♦ss♦♥ ♦♦♠♦♦② ♥ q♥t③t♦♥

♥ ♥ t ♦ ♣s té ♥ ♣s t

❬♦r♦s♥ t ❪ ♦r♦s♥ t ❱ ts♦ ♠② ♦ P♦ss♦♥

strtrs ♦♥ r♠t♥ s②♠♠tr s♣s té ♥ ♣s t

❬♦st♥t ❪ rtr♠ ♦st♥t ♥t③t♦♥ ♥ ♥tr② r♣rs♥tt♦♥ Prt Pr

q♥t③t♦♥ tr ♥ ♠♦r♥ ♥②ss ♥ ♣♣t♦♥ ♣s ♣

té ♥ ♣

❬♦t♦ ❪ ①② ♦t♦ ♠rs ♦♥ ♦♠tr ♥t③t♦♥ ♦ ♠tr① ②♣ P♦ss♦♥

rts ♦rt t ③ té ♥ ♣

❬rss ❪ rss ♦t♥ rt ♥ ♥♦♥ rs t

♦ts t ♥♦ ♣s té ♥ ♣s t

❬♥r♦③ ❪ ♥r♦③ s rétés P♦ss♦♥ t rs èrs s

s♦és ♦♠ ♦ ♣s té ♥ ♣

❬♦t♦ ❪ ♦t♦ rs ♦r♠r ♦r r♥ sss ss♦t t ♦rt♠

♦♥♥t♦♥s ♦②♦ té ♥ ♣

❬t♦ ❪ r♥♦ t♦ ♦ ♣ ♦r ♦♦♥♦♠ ②st♠ P ②♦t♦ ❯♥

♦ ♥♦ ♣s ♠ té ♥ ♣s t

❬P♦s ❪ ♠ P♦s r ♦♠tr② ♦ P♦ss♦♥ rts ♦r♥ ♦

t♠t ♥s ♦ ♥♦ té ♥ ♣s t

❬♥rt ❪ ♥rt r♥t ♦r♠s ♦r ♥r ♦♠♠tt rs r♥s

♠r t ♦ ♦ ♣s té ♥ ♣

❬t♦ ❪ ②♦ t♦ ♦r② ♦ ♦rt♠ r♥t ♦r♠s ♥ ♦rt♠ t♦r

s ❯♥ ♦②♦ ♦ ♣s té ♥

♣s t

♦r♣

❬♦r ❪ ♦r trtr s s②stè♠s ②♥♠qs ♥♦ té ♥

♣

❬r ❱rr ❪ r♠♥♦ r t ♥♦s ❱rr ❱rétés rt

rs s s♦t♦♥s ♣tqs P Pr♦♥s ♦ t ♥♦r♥ ♦♥r♥ ♦♥

♦♠tr② ♦♠② ♥st♥ ♦♦ ♥② ♣s

té ♥ ♣

❬❯r♥ ❪ ❲ ❯r♥ ♣rq♥t③t♦♥ ♣rs♥tt♦♥ ♦ P♦ss♦♥ r

♥ t ♦ ♣s té ♥ ♣s t

❬❱♥♦r♦ ❪ ❱♥♦r♦ ♦ r ♦ ♥r rt ♦♣rt♦rs

♦t t ♦ ♦ ♣s té ♥ ♣

és♠é ♦t tt tès st ♣r♦♣♦sr s rtèrs ♣réq♥tt♦♥ s

strtrs P♦ss♦♥ à s♥rtés ♣♦rtés ♣r ♥ sr r ♥ rété ♦♠♣①

♠♥s♦♥ ♥

P♦r ♥♦s ♣rt♦♥s ♥ ♦♥strt♦♥ érq s ér♥ts ♦r♠s ♦rt

♠qs ♦♥ ♥ é ♥♠♥t ♥♥ré t ♣r♦♣r ♥ èr ♦♠♠tt ♣♦r

♥tr♦r ♥♦t♦♥ èr P♦ss♦♥ ♦rt♠q Ps ♥♦s ♠♦♥tr♦♥s q ts

strtrs P♦ss♦♥ ♥s♥t ♥ ♥♦ ♥r♥t ♦♦♠♦♦q ♣r t ♥

strtr èr ♥rt qs ♥s♥t sr ♠♦ s ér♥ts

♦r♠s ♦rt♠qs râ à r♥r ♥♦s ét♦♥s s ♦♥t♦♥s ♥térté s

ts strtrs P♦ss♦♥

♦t ♦r ♥♦s ♠♦♥tr♦♥s q ♣♣t♦♥ ♠t♦♥♥♥ t♦t strtr P♦ss♦♥

♦rt♠q s ♣r♦♦♥ sr ♠♦ s ér♥ts ♦r♠s ♦rt♠qs t

♥t ♥ strtr èr ♥rt sr r♥r ♣s ♠ tt

♣♣t♦♥ st ♦♥t♥ ♥s ♠♦ s ért♦♥s ♦rt♠qs ♦s ♣♣♦♥s

♦♦♠♦♦ P♦ss♦♥ ♦rt♠q ♦♦♠♦♦ ♥t ♣r tt r♣rés♥tt♦♥

Pr st ♥♦s ♠♦♥tr♦♥s sr qqs ①♠♣s q s r♦♣s ♦♦♠♦♦s

P♦ss♦♥ t ① P♦ss♦♥ ♦rt♠q s♦♥t ♥ é♥ér ér♥ts ♥ qs ♦ï♥♥t

♥s s s strtrs P♦ss♦♥ ♦s②♠♣tqs

♦s tr♠♥♦♥s ♣r ♥ ét s ♦♥t♦♥s ♥térté ts strtrs ♠♦②♥

tt ♦♦♠♦♦

♦ts és trtrs P♦ss♦♥ ♦♦♠♦♦ P♦ss♦♥ sr r

èr ♥rt q♥tt♦♥ ért♦♥ ♦♥trr♥t ♦rt♠q strtr

♦s②♠♣tq strtr P♦ss♦♥ ♦rt♠qs

strt ♠♥ ♦t ♦ ts tss s t♦ ♣r♦♣♦s rtr ♦ ♣rq♥t③t♦♥ ♦

s♥r P♦ss♦♥ strtrs t s♥rts rr ② r s♦r ♦ ♥t ♠♥s♦♥

♦♠♣① ♠♥♦

♦r ts strt r♦♠ ♥ r ♦♥strt♦♥ ♦ ♦r♠ ♦rt♠ r♥ts ♦♥

♥t② ♥rt ♥♦♥ tr ♦ ♦♠♠tt ♥ ♥tr② r ❲ ♥tr♦

t ♦♥♣t ♦ ♦rt♠ P♦ss♦♥ r ♥ s♦ tt ts P♦ss♦♥ strtrs

♥ ♥ ♦♦♠♦♦ ♥r♥t ts s ♦ t ♥rt r strtr

tt t② ♥ ♦♥ t ♠♦ ♦ ♦r♠ ♦rt♠ r♥ts ❲t t ttr

st② t ♥tr ♦♥t♦♥s ♦ s P♦ss♦♥ strtrs

rst s♦ tt t ♠t♦♥♥ ♠♣ ♦ ♦rt♠ P♦ss♦♥ strtr ①t♥s t♦ t

♠♦ ♦ ♦r♠ ♦rt♠ r♥t ♥ ♥s strtr ♦ ♥rt

r ♦♥ t rtr♠♦r s♦ tt ts ♠ s ♦♥t♥ ♥ t ♠♦ ♦ ♦rt♠

rt♦♥s ❲ ♦rt♠ P♦ss♦♥ ♦♦♠♦♦ t ♦♦♠♦♦ ♥ ② ts

r♣rs♥tt♦♥

sq♥t② s♦ ♦♥ s♦♠ ①♠♣s tt P♦ss♦♥ ♦♦♠♦♦s r♦♣s ♥ P♦ss♦♥

♦rt♠ ♦♦♠♦♦s r♦♣s r r♥t ♥ ♥r t♦ t② ♦♥ ♥ t s

♦ ♦s②♠♣t P♦ss♦♥ strtrs ❲ ♦♥ t st② t ♣rq♥t③t♦♥ ♦♥

t♦♥s ♦ s strtrs ② ♠♥s ♦ ts ♦♦♠♦♦②

②♦rs P♦ss♦♥ strtrs P♦ss♦♥ ♦♦♠♦♦② r s♦r ♥rt r

q♥t③t♦♥ ♦s②♠♣t strtr ♦rt♠ P♦ss♦♥ strtrs ♦rt♠ ♦♥trr

♥t rt♦♥