VirtualX.ge#ofQu4sche-s onsurfaced.

O.

Introduction

X smooth projective surface Ici, CE Hak

, 2) e tf (X ,Z ).

1) Fgm ( X,

us GW invariant ⇒ k . theoretic GW

2) MTH ( v ) us VW invariant = ( M ) ⇒ refined VW = Ijf M )EGK

,2017 ]InQuihuis → Quintus,!;ye

-- Eiland) ⇒ ref

Qxjyt! wait

[ op ,2019 ]\

⇐deignPrinciple : Building blocks of sheaf theoretic moduli space on surface X are

ED→ universal formula EEG 42001 ]

Hilbert scheme of g

points X

divisors Hill → SW invariant EDKO,

2007 ]

1.

virtual invariant of Quit scheme

Quirk CET e. n ) : -

- f E - is → of 'Ia→ . ] /RKCQ) -

- o,

c. CQ ) -

- f,

XCX,

Q ) -

- n

}

Tan = Howls.

Q )] difference = v. dim land ) = Nn t {

.Obs = EH'

( s,

Q )

Ots'

= EATS,

Q ) Hom ( Q,s④k×Y = o

⇒ 7- z - term Pot ⇒ [ anti "

.O + .

To:L,

stand - - .

EFG,

20103 evil Quit ) S[ ⇒

in Crd (TIE ) E Z

y-

-

tfrerigour) : =

,cystx cant

,Ari e Zap

go 'tgland)

2.

Work of [ 01320193 on ( Quit ) S Conjecture.

Def. 2¥:{ q ) Iz

,

E E"( Quark en ) E Q KED

.

⇒ explicit formula.

→min

. :÷÷÷::÷:'

z ::* . . J→ X minimal surface ofgeneral type with pg①.

( t simply conn,

⇒ E ¥141)[ lkx ( o see N )

⇒

.ee#re-- E'' ' ' 'HEY'T .¥wfSw ( Kx) ITI -

- N - l.

3) I → X blow up of rational surface, f- E

,N -

- I

⇒ 2¥:.ec#--q . ( III )"

in .

me= 2x

,.

.

subring.

( OP ) X simply connected ⇒ Z elf ) e QCE ) E QUIDmm

i. e. ,

coefficients arerelated

.

Think ) Xany

surface with pg ⇒ meet ) E Q lot ). +

key :

Iecq , e a yxq ,

? ' ".

1)

RefinedQuit scheme invariant I SW invariant

.

e -2 .

X min genotype ,Ps > o ⇒ swlkx ) -

- H" " '

2) Multiplicative universal formula ( under certain condition 8 )

e.g .UN

,AJ

3) Blow up formula ( under ¥ Is blow upterm rational ?

e. a . 2g , ⇐LEI FEI2x

, , HE?

}.

See.bz - Witten invariants of E ' ' ko ]. jwthey / out ahem . theory

.

X smooth pug surface, CE H' H2 )

. ( Mochizuki ) ( virtual localization

Hills : =/ DEX effective divisor sit.

c. ( Gass) =p }.

t some calculation)

Tan = HYO ,>CD ) ) ,

Obs = H' ( ODD ) )

⇒ I a - term pot of vde=RK .

considerg

he Oiiiibxxl D) ftp.wq *

s h = " ( he ).

IAJ : Hills → Pic '

sending Dts QCD ).

Define Sw ( e ) : =

,

A Ix ( hkn-LHilh.IT"

) e H* l pic ) re AH 'Cx. 2)

.

If vde -

- o

,

then AH'

(X. 2) = Z.

4 U

SWIM = dyEHHB.TK

4.

Correct assumption A S Math result.

Det ( L ) Wesay e is swing if

[ te -

- et .. . ten with Sw to ti ) ⇒ [ vdq=o ti ]

.

7- many examples ( all cases in top ] t new )ed . IAI X any

surface, f-

o ⇒ f : Sw length N.

-

: Only effective decomposition : f- Ot . - to f vdq,

-0=0.

N%lB7a :X → C relatively minimal elliott surfacee : su length N

.

-

q . supported on the fibers ( '

" " EF -

- o ). k×n rate mutt of F

Zariski Lemma

II Xany

surface withpg > o⇒it

e : Sw length N.

ED Ko ]I

⑤ f : Sw length N → file : SW length N.

Mainthm ( L ) Assume e : Sw length N.

7- universal seise UN,

Umi,

W,

E QLY.eu '

. - . ie

" ) KED s.tn.

The generating series of equivariant refined Quit scheme invariants is

z÷÷ian⇒=E÷÷÷÷¥.su#uii.x..niiis:.x..uwiii:.

run'

f nuttily's Tx " " 's.

Rink.

7- similar formula for monopole contribution to ref VW invariants :

÷:÷:::÷÷÷÷÷÷÷:÷÷÷i÷:⇒÷÷known !

( Sketch of proof of

mahthnqswkyt

In general , geometry of higher N Qut scheme is poorly understood.

Let E' no EN with distinct weights w.

,

. war.

⇒ E' a QWEN.eu )

( similar as a IP" )

( Quote ,

= UQuoted

, ein Dx .. . x Qnrkce

'

, inn ).

f- fit . . t fun = hit . . t HN

Induced obstruction theory of each fixed loans!

pits into factors.

[ GSY,

2017 ] virtual class ofQuestion : what is [ Quoted

, Eh 55"

? nested Hilbert scheme.

7- identification Quoted.cn?--XtmkHilbe--X "{ here m=n+

LF,

1968 )2

.

Propels Eantxcesensj"

[ X' I"

e ( coke'

"

) n #7 XEHH )-

rank m

pf, ① : compare .② Special case of [ GT

,2019 ]

By virtual localization of [ GP,

1999 ],

surfs : ) 's.

x canteen.

* *fwineskin

--

- 2 swigs . - Sw 2 J C sth )q= ft

. . . tf n-

- hit . ' thx ×EnP× . ..×XTmN )

fHills" ] safe; ) ⇒

f sit. vdq.

-

- o

( sth )⇒ zi÷÷i"÷÷÷÷" " " " 'n .xi⇒ .

# E EGL,

2001 ]

7- multiplicative universal formula.

THE

5. Applications of the Main -1hm

.

( to each case EINE ).

ThmA_.

Xany

surface,

9=0 .

( z in -

- on"

where

=4-E) 4 - Yat )

ITIN UNL,

= . . . woo yyuqyN ⇒( I - l HY ) ti t Ytitj ) e 4) If )

.

run

Here t .

.

.. . .tn are rots of

of-

- IN.

.

.

.

'

EPN:polynomial in y , of .

e.g ,

. P ,= I

. P .-

- I - utiytyyq + yzqr* Putty ) = ly'VE )

" !pncqt , y

-

y.

. P ,= I - ( Hay -19 y 't y

') ft ( Italy -136-1458/+36 y4t9y5tyb ) E

- ( ztaytay 't y'

E t y' ft .

By taking y-

- I,

it recovers Top ].

Thmts.

a :X → C relatively minimal elliptic surface, f

: supported on the fibers.

( ziiiii "

t.ei.i.iii.is" "

.

By [ DKO ],

SW invariantsabove are computed by

Shoki ) = I c .

Dd (22-21×10×3)E- dFt§ajFj

St.

d > o

,

o Eajcmj

where F 's are multiple fibers of multiplicity ng .

Time.

X minimal surface of general type with pg > o.

For

any f- lkx ( OELEN ),

we have GN.e.GE QYVE '.

( z :"

:* : "-

. E " "

.com#e. uhm

a.eu#.i;Eiix.I, x .¥mI÷i¥÷⇐.

( Set s -

- N - l

Note that swlkx ) = HI ' " )by [ Ck

,2013 ]

.

By taking y-

- I,

we recover Cop ] S lift technical assumptions

( simply conn,

I EHhelkx 'D

Thnx.

Xany

surface. f : SW length N

.

( Blow up formula)

( ÷::::" " " " " " " " "

.

Ii !, econ

-

- ( E. Bine) .

" got ,where

Blye = -2 As e Qlyllf ).

JE EN ]

IJI -- N - l

e. 2 . 131µm -

- I,

131µm,= .

445-4-554( I -

of ) C I - y 'VE ) .

Forany given N

, by ,it is

easyto compute Greg ,

Blye.

I

2=0 .

[ Thin

A⇒ [ µ of ) c- Qcyslq ) if X is surface with

pg > o ).

(proof ) Let X be surface with pg > o.

Then

bfmhapply b/c Ps > o

.

since Bbv,

ee Qty) I of ),

we mayassume X :

minsurface

.

o kod -

- o : K3 or abelian ⇒ f- o is only SW class ⇒ Thin IAI.

( o kod -

. I :i⇒ I ⇒ time. )

° lad = z : minimal surface of ⇒ f- Kx is only SW class ⇒ Than II.

general type withpg > o

TEH

Question : what happen ifIEE?

[ Jop ,2020 ] X simply connected surface

.

Then 2 E) EQCE).

-

In fact, they study more general descendents

.

6.

Reduced invariant of k3 surfaces.

X k3 surface.

Usual Quit scheme invariants are trivial Kc

⇒ surjective

cosetteObs = Exits .Q ) Is EHIQQ ) H' tox ) a E

.

I I.

: Ext't 0,970-3=0 i

: Hom I Q ,Q④kx ) ← Hock )

:. We consider reduced obstruction theory with

= kerf Obs → E).

L ) X k3 surface,e:ptgfN .

⇒ xjdcaukcd.e.us ) = x.glEI)where E → IP is unheard curve in class f S m = ntlg - I )

.

Rmd.

D Above thin was proven in [ op ) on the level of Euler characteristic.

2) CENT isproven

to be smooth S ht'

' FLEET ) are computed by [ KY,

2000 ].

F - -

Together with Thu t KY formula

Core.

with

privationI !

, .eu ,= I tix ! canteens ) is my

coefficient of fed in

ftp.fy .

a-EY 'll - Ey ) a - Elysa - EFI) a - Etty ) h - Etf ) 4- Easy ) .

↳ Kawai - Yoshioka.

In particular,

I" tie

, is rat 't in tradable.

=

7.

Further questions

"Qut scheme invariants

Lwinvariant

stableinvariant

→X with ⇒ rationality gnestfn ?3.)Reda variant of k3 surface with higher N / ( non - primitive.

- - me(Define non - trivial reduced invariant of ambling.4.)Relative they Quit,yp( v ) S degeneration formula

.

-

(

-