Hodge-theory:

Abel to Deligne

Phillip Griffiths

Talk given at the IAS, October 14, 2013.

I this will be mostly an historical talk — it is not intendedas an overview nor a current state of affairs of Hodgetheory — it will also be informal — intent is to give asense of things and how they fit together, not to presentprecise statements

I the central point of the talk will be the analogyI Abel took a subject (elliptic integrals) in which there

were very interesting special cases, and set it in a newand general context that opened up the subject

I similarly, Deligne’s theory of mixed Hodge structurestook the subject of classical Hodge theory for smooth,projective varieties and recast it for arbitrary complexvarieties and maps between them, thus similarly openingup the subject of Hodge theory

I will begin by recounting Abel’s theorem in its originalform and then trace some history, emphasizing especiallythe work of Picard which is perhaps less familiar, upthrough Lefschetz, Hodge and Deligne’s theory,concluding with some comments on the recent work ofM. Saito who took mixed Hodge theory and variations ofHodge structure and fused them into the currentlypenultimate state of the subject

I my presentation of the historical development issubjective and personal and does not reflect a researchedaccount of the development of the subject — I will alsodraw on some recollections of what was told to me as astudent by Lefschetz, Kodaira and Spencer, and later byHodge when I visited him in Cambridge (UK)

I finally, as we all know there are issues about what was bycurrent standards actually proved by the classicalmathematicians — here I will draw three levels ofdistinction

I an argument that can readily be made into a modernproof (e.g., Abel’s theorem or Picard’s version of thealgebraic de Rham theorem)

I an argument that is at least to me convincing but I havenot written out all the details (the existence of thePicard variety of ∞q rationally inequivalent curves on analgebraic surface S , where q = h0(Ω1

S))I an argument that is incomplete but which was later done

(e.g., Lefschetz’s for the hard Lefschetz theorem andHodge’s for the Hodge theorem)

〈〉

Abel

I 18th century mathematicians were interested in integralsof algebraic functions

ˆr(x , y(x))dx , f (x , y(x)) = 0;

they arise in geometry

ˆ √dx2 + dy 2 =

ˆdx√

1− x2for x2 + y 2 = 1

and in mechanicsˆ √p(x)dx for y(x)2 = p(x)

I of particular interest were elliptic integrals

x2/a2 + y 2/b2 = 1, a > bˆ √a2 cos2 θ + b2 sin2 θdθ = a

ˆ(1− k2 sin2 θ)dθ√

1− k2 sin2 θ

= a

ˆ(1− k2x2)dx√

(1− x2)(1− k2x2)

where k2 = (a2 − b2)/a2 and x = sin θ; by analogy withthe formulas for sin 2θ, cos 2θ for doubling the arc lengthof a circle, the mathematicians of that period (CountFagnano, Euler, Lagrange, Gauss, . . . ) found formulas fordoubling the arc length of the ellipse;

I also much studied was the arclength of lemniscate =locus of a point p = (x , y) such that product of distancesfrom two fixed points p1, p2 is constant

r r rr

H

HHHHHHH

p2=(−a,0) (0,0) p1=(a,0)

(x ,y)

r rp2 p1

ˆ √dx2 + dy 2 =

ˆdx√

1− x4

Here we have taken 2a2 = 1

I The integral above was understood as the “function” ofthe upper limit ξ of integration and by choosing a path γof integration from x0 to ξ together with a branch of themulti-valued function y(x) along γ

F (ξ) =

ˆ ξ

x0

r(x , y(x))dx

x0

s γ

∗∗

sξ∗ = branch points∗

A different choice of path changes F (ξ) by a period(which is where we now think of as Hodge theory started).When f (x , y) ∈ Q[x , y ] and ξ ∈ Q the transcendence ofF (ξ), as well as of the periods, is a much studied andvery beautiful subject (Siegel, Bombieri, etc.)

I when deg f = 1 such integrals of algebraic functions maybe expressed in terms of elementary functions (partialfractions), and when deg f = 2 they may also be explicitlyevaluated, essentially because conics are rational curves

(x(t), y(t))

t

but when deg f = 3, e.g., f (x , y) = y 2 − p(x) wheredeg p(x) = 3, these integrals were of a different character;

I as noted above, there were a number of specific resultsabout elliptic integrals when Abel recast the wholesubject (1820) — he introduced two general concepts

I abelian sumsI inversion

for the first he considered the curve C = f (x , y) = 0together with a family of curves Dt = g(x , y , t) = 0depending rationally on a parameter t —

Dt

C

f (x , y) = y 2 − p3(x)

g(x , y , t) = ax + by − t

and setting

C · Dt =∑i

(xi(t), yi(t))

he considered the abelian sum

u(t) =∑i

ˆ xi (t),yi (t)

(x0,y0)

r(x , y(x))dx

Theoremu(t) is an elementary function; i.e., u′(t) ∈ Ct

This is a very general result; the only implicit assumptionis that f (x , y) has no multiple factors

Translated into a current framework, Abel’s proof was thefollowing:

I

p

q

6666666666= (x , y , t) : f (x , y) = 0,

g(x , y , t) = 0 = incidence variety;

C P1 = t-line

for ω = r(x , y)dx |C , the trace

Trω = q∗p∗ω

is a rational 1-form on P1 — by local considerations it iseverywhere meromorphic (no essential singularity), andhence is rational —

u(t) =

ˆ t

t0

Trω;

I when deg f = 3, a new phenomenon arises — setting

ω =p(x , y)dx

fy (x , y)

∣∣∣∣C

deg p 5 n − 3⇒ u(t) = constant;

the point is that deg p 5 n − 3⇒ Trω is holomorphic onP1;

NoteThe particular form arises from

ResC

(p(x , y)dx ∧ dy

f (x , y)

)=

p(x , y)dx

fy (x , y)

∣∣∣∣C

I for inversion, if we define x(u), y(u) by

u =

ˆ (x(u),y(u))

(x0,y0)

ω

then Abel’s theorem translates into an addition theoremfor the x(u) and y(u) — e.g. for the Weierstrass cubic,and choosing (x0, y0) properly, Abel’s theorem is

u1 + u2 + u3 = 0;

which givesx(−(u1 + u2)) = R(x(u1), x(u2), y(u1), y(u2))

y(−(u1 + u2)) = S(x(u1), x(u2), y(u1), y(u2));

using fy = 2y and the fundamental theorem of calculus

dx

2y

∣∣∣∣C

= du

which for the Weierstrass p-function x(u) and itsderivative y(u) gives the usual addition theorems; Abelviewed his result in part as a very general additiontheorem;

I an important point is that C may be singular —essentially and in hindsight Abel was using the mixedHodge structure on C to describe what we now call thegeneralized Jacobian — for C smooth of degree n

H0(Ω1C ) ∼= polynomials of degree n − 3 ∼= C

(n−1)(n−2)2 ,

and for C possibly singular the RHS gives the F 1 for thelimiting mixed Hodge structure as a smooth Cs specializesas s → 0 to C = C0;

I Abel’s theorem opened the gateway to the modern theoryof Hodge structures of weight one associated to compactRiemann surfaces — Jacobi, who formulated the Jacobianvariety

J(C ) = H0(Ω1C )∗/periods

and proved the inversion theorem and a converse toAbel’s theorem

I In more detail, note that when C is smooth it is complexsubmanifold of P2 and hence is a compact Riemannsurface with space H0(Ω1

C ) of holomorphic differentials —the periods are

ˆγ

ω, ω ∈ H0(Ω1C ) and γ ∈ H1(C ,Z);

choosing a base point p0 ∈ C , abelian sums correspond to

d∑i=1

ˆ pi

p0

ω modulo periods,

p1

p0

p2

it is the periods that correspond to what is usuallythought of as Hodge theory.The object J(C ) is a complex torus and the geometry ofthe above map

C (d) → J(C )

was the main interest of classical theory.

— Riemann, who conceived Riemann surfaces andshowed that

I h0(Ω1C ) =

(12

)b1(C )

I H1(C ,C) ∼= H0(Ω1C )⊕ H0(Ω1

C )

= H(1,0)(C )⊕ H0,1(C ),

although these mathematicans of the 19th century wereable to work with some singular curves as necessary, therewas no general formulation of the intrinsic Hodge theoryand of the relation of the Hodge theory of a generalsmooth Cs to the Hodge theory of a specialization — forthis we had to wait for Deligne, whose worked opened theway for Schmid, Clemens-Schmid, Steenbrink, andCattani-Kaplan-Schmid, . . .

Picard

I before discussing him I want to say something aboutLefschetz — he was born in Moscow (1884) and familymoved to France — went to school there and lovedmathematics but studied engineering because he thoughtthat, not being French, it would be hard for him to get anacademic position there — came to the US to getpractical experience and lost both of his hands in a boileraccident at a foundry in St. Louis — so enrolled ingraduate school in mathematics at Worcester Polytechnicwhere there were three faculty, one of whom worked onthe classical theory of plane curves (how many doublepoints and cusps can an irreducible curve have?)

— took a job at the University of Kansas where he wasfor 10 years and did his work in the topology of algebraicvarieties, and also his fixed point theorem in topology —he took with him the works of Picard and wrote “theseprovided the wellspring for me” (he also told me that hewas not allowed to teach introductory complex analysisthere because they didn’t think he knew the subject wellenough)

I Picard studied algebraic surfaces, which for him weregiven by

S = f (x , y , z) = 0 ⊂ P3,

a surface with ordinary singularities in P3 — he wanted toextend the study of integrals of the 1st, 2nd and 3rd kindon a curve to this case — for him the method was tofibre the surface by curves

Cy = f (x , y , z) = 0, y = constant ;

in a modern framework

Sπ

=

desingularization of S

blown up along thebase locus of the pencil

P1

I he introduced the picture

ssyN

s sPP

PPPs

bbbbb

"""""

y0

y1

where Cy0 is a reference curve and the Cyi have a singlenode —

I the complement of the above configuration is contractiblein P1 = Riemann sphere, and so the whole surfaceretracts onto the part lying over the slits in the complexplane.

I for the node

analytically: u2 = v 2 − t u =√

v 2 − t−√t

√t

topologically

locally the topology isδ

γ

I thus we may topologically picture the surface assomething like

Cy0

I the key to understanding this is the monodromy action ofπ1(P1\y1, . . . , yN) on H1(Cy0 ,Z);

I for this the first step is the local monodromy around eachof the singular curves;

Picard drew the picture below and wrote“as t turns around the origin we get for γ”

∼

Lefschetz said that he spent hours staring at that picture— Picard first seems to have arrived at it analytically

1

2πi

ˆγ

du

v= log t + holomorphic function of t;

i.e.d

dt

(ˆdu

v

)=

1

t+ h(t)

I this gives the local monodromy

γ

δ

γδ δ → p

p

δ → δ

γ → γ + δ

(Picard-Lefschetz transormation);

I the cycles in H1(Cy0) were divided intoH1(Cy0)ev = vanishing cycles

H1(Cy0)inv = invariant cycles

I the global monodromy is another story that we will cometo;

I Turning to differentials on the surface, Picard studied

1-forms ϕ =p(x , y , z)dx

fz(x , y , z)+

q(x , y , z)dy

fz(x , y , z)

2-forms ω =r(x , y , z)dx ∧ dy

fz(x , y , z),

for the holomorphic 1-forms he divided them into twogroups

ϕ1, . . . , ϕq︸ ︷︷ ︸ ϕq+1, . . . , ϕg︸ ︷︷ ︸that varied holomorphically in y ∈ P1 — in modernlanguage

R1πωS/P1

∼=q⊕

OR1 ⊕

(q⊕

i=q+1

OP1(ki)

)

where all ki = 1 — Lefschetz identified those whereki = 2 as being H0(Ω2

S) under ϕ→ ϕ ∧ dy – will returnto this later, all of the above is correct

I on the topological side, Picard also gave a correct proofthat

H1(Cy0 ,Z)→ H1(S ,Z)→ 0

H2(S ,Cy0 ,Z) is generated by the ∆i =

locus of δialong the

segment y0yi

=

PPP ;

here things get a little murkey as he essentially assertsthat

H1(Cy0 ,Q)ev ⊕ H1(Cy0 ,Q)inv = H1(Cy0 ,Q)

dim H1(Cy0 ,Q)inv = 2q;

where “ev” = vanishing and “inv” = invariant

in any case we know this as a consequence of regularHodge theory, and the vast generalization of the abovedirect sum is a consequence of the semi-simplicity ofglobal monodromy (theorem of the fixed part) whichresults from Deligne’s mixed Hodge theory;

I he did, together with Poincare, give a correct proof that

“there exist ∞q rationally inequivalent curves on S”;

this is an existence theorem, whose proof was a step inLefschetz’ proof of his famous (1, 1) theorem — if timepermits will sketch this argument, now much extendedand recast by Saito and others;

I one of the interests was to classify rational differentials onsurfaces extending the situation for curves as

I first kind = holomorphic differentialsI second kind = differentials without residues (and thus

may give cohomology classes)I third kind = everything else

I the classification of 1-forms was a fairly direct extensionof the case of curves

I turning to the 2-forms, Picard gives a correct proof of theaffine algebraic de Rham theorem

H2(S0,C) ∼= H2DR

(Γ(Ω•

S0,alg), d)

(he did this also for 1-forms) — for 2-forms he found thenew phenomenon that there were ω which areholomorphic on S0 and where

ω = dψ

with ψ rational on S but (ψ)∞ % (ω)∞; this cannothappen for 1-forms

in fact, the number of such relations was equal to ρ− 1where, in modern language

ρ = dimHg1(S) = rankNS(S)⊗Q

— this focused attention on the homology classes carriedby algebraic cycles — Picard’s statement was

dim space of differentials of the 2nd kind/exact = b2(S)−ρ

— all the classical results of this kind are now“explained” using mixed Hodge structures and the variousexact sequences that result from that theory;

I As for Lefschetz, who wrote that he “put the harpoon oftopology into the whale of algebraic geometry,” I will notsay much as his basic results on the topology of algebraicvarieties are well known — will comment on the so-called“hard Lefschetz theorem” — in the above discussion ofalgebraic surfaces, for an invariant 1-cycle λ we let

Λ = 3-cycle on S traced out by λΛ · Cy0 = λ;

this is defined as class in H3(S ,Q), and every class therearises in this way; hard-Lefschetz is the statement that

∩Cy0 : H3(S ,Q)→ H1(S ,Q)

is injective (and must then be an isomorphism byPoincare duality) — this is the same as

H1(Cy0 ,Q)ev ∩ H1(Cy0 ,Q)inv = (0)

just as with the issue discussed above, it seems thatLefschetz’ argument for hard Lefschetz is incomplete (andthere are philosophical reasons why this should be thecase) — Lefschetz had of course incredible geometricintuition, and it was said of him that he never stated afalse theorem nor gave a correct proof — when I was agraduate student he once gave a talk in the old Fine Hallon the topology of algebraic varieties and after giving anintuitive “cycle traced out by the locus of. . . ” argumentone of the young turks in the audience objected that thiswas not a proof, to which Lefschetz replied “come now,we’re not babies after all”

— the point though is that hard Lefschetz remainedunproved until Hodge — moreover, in general the cycletraced out by the locus of the invariant cycle λ closes uponly if d2λ = 0 in a spectral sequence, which may not bethe case unless we are in Kahler case;

I Hodge did his work in the 1930’s and the stated mainpurpose of his book “Harmonic integrals” was to prove,using differential forms and the recent de Rham theorem,Lefschetz’s results on the topology of algebraic varieties— he had as a model Hermann Weyl’s book “Die Ideeder Riemannaschen Flachen,” in which he formulated into this day very modern terms Riemann’s results on theexistence of a polarized Hodge structure

H1(C ,C) = H1,0(C )⊕ H0,1(C ), H0,1(C ) = H1,0(C )Q(H1,0(C ),H1,0(C )) = 0

iQ(H1,0(C ),H1,0(C )) > 0

where C = compact Riemann surface, H1,0(C ) = H0(Ω1C )

and Q = cup product (intersection form via Poincareduality) using harmonic theory — minimize

‖ϕ‖2 =

ˆC

ϕ ∧ ∗ϕ, ϕ ∈ A1(C )

in its cohomology class — Hodge sought to do the samein general, the difference being that in higher dimensionsa Riemannian metric was needed to define ‖ϕ‖2;

I Hodge’s work fell into two parts:

A. HqDR(X ) ∼= Hq(X ) = harmonic q-forms where (X , g) =

compact Riemannian manifoldB. the special structure of Hq(X ) that arises when (X , g)

is a Kahler manifold

I will mainly discuss B, but will note that A has aninteresting history involving the IAS. Namely, theimportance of Hodge’s work was of course recognized andwas the subject of a seminar conducted here by Weyl —the sense at the time seems to have been that Hodge’stheorem A was almost certainly true, but his proof wasincomplete

— Hodge’s book was before the modern theory of linearelliptic PDE’s, and it was not clear that (i) a minimumexists in L2, and (ii) even if it does it may not be C∞ —in putting together a complete argument Weyl devisedthe famous Weyl lemma

ϕ ∈ L2 and (∆ψ, ϕ) = 0 for all C∞ψ ⇒ ϕ is C∞

(weak solutions of linear elliptic equations are smooth)

I part B was done by calculations, using classical tensorcalculus notations (indices galore), and was a major tourde force — the two main points are

[∆, πp,q] = 0

where ∆ = Laplacian or differential forms and πp,q =projection of a differential form

ϕ =∑I ,J

ϕI ,Jdz I ∧ dzJ

onto the terms where |I | = p, |J | = q; and

[ω,∆] = 0

where ω =(√−12

)∑i ,j gi jdz i ∧ dz j is the Kahler form —

both of these were lengthy, very difficult computations

— the first of these has at least been put into aconceptual framework by Chern — as for the second, itwas and still is something of a miracle, as in general

∆(ϕ ∧ ψ) = ∆ϕ ∧ ψ + ϕ ∧∆ψ + big mess;

Hodge told me that he proved it because it was the onlyway to get Lefschetz’s results to come out; (I have notchecked this, but my guess is that for a positive (1, 1)form ϕ =

(i2

)∑gαβdzα ∧ dzβ, the “big mess”

= 0 ⇐⇒ dϕ = 0)

I in any case one has the Kahler package

(i) H r (X ,C) = ⊕p+q=r

Hp,q(X ), Hq,p(X ) = Hp,q(X )

(Hodge decomposition giving a Hodge structure ofweight r)

(ii) Lk ,Hn+k(X )∼−→ Hn−k(X ), dim X = n (hard Lefschetz)

(iii) H r+k(X )prim =: kerLk+1 : Hn+k(X )→ Hn−k−2(X )and

H r (X ) = ⊕s=0

LsH r+2s(X )prim

(iv) Q : Hn+k(X )prim ⊗ Hn+k(X )prim → CQ(ϕ,ψ) =

´X Lkϕ ∧ ψ gives a PHS,

where L = cup product by the Kahler form ω

I note that the polarizations depend on the choice of[ω] ∈ H2(X ,R) — when [ω] is rational (think ofX ⊂ PN∗ and [ω] = hyperplane section), (ii)–(iv) aredefined over Q — for later use, we note two points

I the first is that the Hodge decomposition (i) isequivalent to giving Hodge filtrationF r ⊃ F r−1 ⊃ · · · ⊃ F 0 = H r (X ,C) where for each p

F p ⊕ Fr−p+1 ∼−→ H r (X ,C),

the relation to (i) isHp,q(X ) = F p ∩ F

q

F p = ⊕p′=pq

Hp′,q(X )

(think of the “p” in F p as “at least p dzα’s in thedifferential form representing the cohomology class)

I the second is that it is the existence of a polarizationthat to a large extent distinguishes the Hodge structuresof geometric interest — e.g., it leads to the very strongcurvature properties of the space of all Hodge structureson a Q-vector space with a given polarizing form Q andHodge numbers hp,q;

I in the 1950’s, Kodaira and Spencer (here at the IAS andlater at the university) built on Hodge’s work, amongother things that resulted were the

I Kodaira vanishing theorem

Hq(X , L∗) = 0 for q < n

where L→ X is a positive (= ample) line bundle

and its equivalence in the Akizuki-Nakano formulationand in case L = [Y ] where Y = smooth hypersurface, tothe Lefschetz theorems over Q about H∗(X )→ H∗(Y ),thereby established a connection between vanishingtheorems and Hodge theory

I in more detail,

Hq(X ,Ωp

X ⊗ L∗)

= 0 for p + q < n

⇐⇒

Hk(X )

∼−→ Hk(Y ), k 5 n − 2

Hn−1(X ) → Hn−1(X )

(usual KVT is the p = 0 case)

I proof thatPic(X ) ∼= H1(X ,O∗Y ) (= Lefschetz (1,1) theorem)

Pic0(X ) ∼= H1(X ,OX )/H1(X ,Z)

where Pic(X ) = CH1(X ) =(

divisors modulorational equivalence D = (f )

);

this gives ∞q rationally inequivalent curves on a surface

I in the mid 1960’s a few fragments of Hodge theory forsingular varieties were around on an ad hoc basis

I generalized Jacobians of algebraic curves (return toAbel)

I Hodge theory for the singular fibres in a Lefschetz pencil,and generalized intermediate Jacobians in this case; atleast to my knowledge there was no concept of afunctorial Hodge theory for all varieties — as with theearlier attempts by Picard, Poincare, Lefschetz, thepresence of singularities created major difficulties inattempts to develop a satisfactory theory of integrals ofthe 1st, 2nd and 3rd kind — then came

I Hironaka’s resolution of singularitiesI Deligne’s mixed Hodge theory

it is interesting to note that in their paper on integrals ofthe 3rd kind, Atiyah and Hodge used resolution ofsingularities to define the log complex

Ω•(log D), quasi-isomorphic to O⊗ Λ•(

dz1

z1, . . . ,

dzkzk

)where D = z1 · · · zk = 0 — as pointed out byGrothendieck, this contained the kernel of the idea for hisalgebraic de Rham theorem, which together withDeligne’s mixed Hodge theory effectively contains thegeneral story of integrals of the 1st, 2nd and 3rd kinds

Deligne

I A mixed Hodge structure (MHS) is given by the data(H ,W•,F

•) whereI H is a Q-vector spaceI W• is an increasing filtration of HI F • is a decreasing filtration of HC

which satisfies

F p ∩Wk,C + Wk−1,C/Wk−1,C induces a pure

Hodge structure of weight k on Grk = Wk/Wk−1;

a mixed Hodge structure is thus successive extensions, ofa special type, of pure Hodge structures — MHS’s admitthe usual operations of linear algebra, and the notion of amorphism between them is the natural one — they alsohave subtle linear algebraic properties, one of which is

strictness: Given ϕ : H → H ′ with ϕ(Wm) ⊂W ′m, ϕ(F p) ⊂ F

′p,then

ϕ(H) ∩ F′p = ϕ(F p)

this implies that MHS’s form an abelian category

Deligne’s TheoremFor X a complex algebraic variety, H r (X ,Q) has a functorialMHS

I for X complete, the weight filtration is W0 ⊂ · · · ⊂ Wr

I for X smooth and open the weights are Wr ⊂ · · · ⊂ W2r

Given a particular presentation of an algebraic variety Xin terms of smooth ones with normal crossingsingularities, it is plausible that there is lurking in H∗(X )a description as some sort of iterated extensions of pureHodge structures. Remarkable to me is the formulation ofthe extension data and, especially, the independence ofthe particular resolution of singularities.

It is safe to say that this result, together with what I willnext mention, opened up the study of the cohomology ofcomplex algebraic varieties in a way similar to whatfollowed from Abel’s theorem for integrals of algebraicfunctions

I Regarding the weight in the weight filtration, veryapproximately the weight of a class ϕ reflects the degreeof the cohomology class on a smooth variety in thestratification that traces through the various exactsequences to give ϕ

I a special type of MHS arises when W• = W•(N) isassociated to a nilpotent endomorphism

N : H → H , N r+1 = 0

then there is a unique W•(N) = W−r (N) ⊂ · · · ⊂ Wr (N),centered at 0, where

N : Wl(N)→ Wl−2(N)

Nk ,Grk∼−→ Gr−k ;

think of the L in the Lefschetz decomposition, only goingthe other way and with a shift in indices— a limitingmixed Hodge structure (H ,W•(N),F •) is one where

N(F p) ⊂ F p−1,

here omitting the polarization conditions

they arise when you have a family

Xπ−→ ∆

with Xt = π−1(t) smooth for t 6= 0 — Deligneconjectured and Schmid proved that the Hodge filtrationson F pH r (Xt ,C) have a limit as t → 0 (this is somewhatsubtle, as we need to take the limit of filtrations on avariable vector space) and the result is a LMHS;

more precisely there is a monodromy transformation

T : H r (Xt0 ,Q)→ H r (Xt0 ,Q)

where(T l − I )r+1 = 0

(quasi-unipotency)

replacing t by t l we may assume that T is unipotent withN = log T nilpotent — then by Schmid, for l(t) =(

12πi

)log t, e−l(t)N · F •t is single-valued and has a limit

F •lim as t → 0; moreover, e l(t)·N · F •lim exponentiallyapproximates F •t .

I Schmid proved his result for any variation of Hodgestructure over ∆∗ — the geometric result was later doneby Steenbrink — in practice, this result tells us how thetopology and Hodge structure behave in a 1-parameterfamily of varieties acquiring arbitrary singularities — theseveral variable version of how the Hodge structuredegenerates was done by Cattani-Kaplan-Schmid withsignificant input by Deligne and continues to play a majorrole in current Hodge theory

I One of the first applications of Deligne’s theorem dealtwith a global family

Xπ−→ S

of smooth projective varieties Xs — here both X and Sare quasi-projective — the basic invariant of this situationis the monodromy representation

ρ : π1(S , s0)→ AutH r (Xs0 ,Z)

with imageΓ ⊂ Aut (H r (Xs0 ,Z),Q)

where Q is the polarizing form on H r (Xs0 ,Q)

— whereas the local monodromies around the branches ofS\S are quasi-unipotent, Deligne’s result is that oppositeis true globally — loosely stated it is that

Γ is semi-simple /Q

this is a vast generalization of the issue that Picard andLefschetz encountered for Lefschetz pencils and was inthat particular case finally settled by Hodge

— will conclude with two further topicsI brief mention of Morihiko Saito’s theory of mixed Hodge

modules, which is the current penultimate version ofHodge theory

I some discussions of existence issues in Hodgetheory/complex algebraic geometry

I in the early 80’s Beilinson-Bernstein-Deligne-Gabberinitiated the development of what is now called theHodge theory of maps

(∗) f : X→ S

where (to keep things simple) f is proper — if there areno singular fibres then Blanchard-Deligne proved that theLeray spectral sequence degenerates at E2 — when thereare singular fibres and dim S = 2 the issue is much moresubtle

— here again the IAS played a role in that a criticalingredient is the intersection cohomology of a completesingular variety, which was created by Mark Goresky andBob MacPherson and which carries a pure Hodgestructure — the theory for (∗) as developed by BBDGused characteristic-p methods (Frobenius acting on l-adiccohomology) — Saito’s stated objective was to put (∗) ina purely Hodge-theoretic setting — the basic buildingblocks of his theory are

I VHS’s and admissible VMHS’s (everything polarizable)I D-modules and perverse sheaves (the latter being

sheaf-theoretic version of intersection cohomology)

very informally, on has the descriptionI Hodge modules are the smallest category of objects over

smooth varieties that contains generically (i.e., on thecomplement of a divisor) polarizable VHS’s and is closedunder direct images

I mixed Hodge modules (the penultimate objects) are thesame with admissible VMHS’s

I application of his theory includeI a general vanishing theorem for the graded pieces of the

de Rham complex of a mixed Hodge module. If (M,F )underlies a mixed Hodge module on a projective varietyX , and if L is an ample line bundle on X , then thehypercohomology of the complex

L⊗[grFp M→ Ω1

X ⊗ grFp+1 M→ · · · → ΩdimXX ⊗ grFp+dimX M

]

vanish in degrees greater than dim X . This resultincludes most of the standard vanishing theorems inalgebraic geometry (such as Kodaira’s theorem) asspecial cases

I Kollar showed that for a projective morphismf : X → Y , with X smooth, the higher direct imagesheaves Rk f∗ωX/Y of the relative canonical bundle aretorsion-free, and vanish above dim X − dim Y . Heconjectured that the same should be true for sheaves ofthe form Rk f∗(ωX/Y ⊗ F pH), where H is a polarizablevariation of Hodge structure. Saito proved this using thistheory.

I On a smooth projective variety X , any set of the formL ∈ Pic0(X ) | hi (X , ωX ⊗ L) = m

is a translate of an abelian subvariety of Pic0(X ), andthe generic vanishing theorem of Green-Lazarsfeld givesa bound for its codimension. This, and many relatedresults, are in fact special cases of general theoremsabout mixed Hodge modules on abelian varieties.

I As a concluding application to illustrate how Hodgetheory and mixed Hodge theory have developed beyondtheir original algebro-geometric roots, I will mention aconjecture/work in progress of Schmid-Villonen:

— Suppose G is a real semi-simple Lie group with Liealgebra g and a maximal compact subgroup K . Theadmissible (g ,K ) modules are certain nicely behavedbimodules for g and K , which were classified a long timeago by Langlands and later reclassified in terms of twistedD-modules on generalized flag varieties by Beilinson andBernstein. The important problem of deciding which(g ,K ) modules are unitary is, however, in some senseopen. There is an algorithm for deciding this question dueto Vogan, however, it is complicated. (Think of defining amodule A as an irreducible sub-quotient of a module B ,which you know concretely, that has a filtration that youknow exists but don’t know concretely)

An important ingredient in making this algorithm feasibleis a certain hermitian form on a (g ,K ) module V . In theessential cases, it always exists and the problem ofdeciding if V is unitary can be boiled down to a problemof the signature of the form on the (usually infinitedimensional) V .

Schmid and Villonen do the following:

(1) interpret the form as a polarization on a mixed Hodgemodule M on a homogeneous space for G associated to V

(2) conjecture that the polarization satisfies sign identities,with respect to the Hodge filtration that M induces on V ,completely analogous to the Riemann bilinear relations ona polarized Hodge structure

This setup has nothing to do with the Hodge theory arisingfrom a family of algebraic varieties and illustrates the extent towhich formal Hodge theory has developed as a subject in itsown right

I finally want to discuss existence issues — e.g.,conjectures of Hodge and Bloch-Beilinson — methods forproving existence include

I deformation theory — works in some cases — existenceof Picard variety either by analytic methods(Kodaira-Spencer) or algebraic ones (Grothendieck) – sofar not in others — e.g., for S a surface andZ (S) = 0-cycles on S , the formal tangent spaces (notdefined here) T f has the property

pg = 0⇒ T f Z (S) = T f Z (S)rat,

but attempts to “integrate” this fall short;

I PDE — linear theory works well (Hodge theorem) —also non-linear theory (Yau, Donaldson-Yau-Uhlenbeck,Simpson, Tian, . . . ) — but geometric theory usingcurrents again falls short — for ξ ∈ Hgq(X ,Z) andY = degree d 0 linear section let

Z = minimal positive integral current with [Z ] = ξ + [Y ];

then Z exists non-uniquely and according to BlaineLawson should be a complex analytic subvariety — oneessential problem is “what does d 0 mean?” Examplesdue to Janos Kollar and others, and recently studied byClaire Voisin, show that there are integral classes

ξ ∈ H2p(X ,Z)

ξ = [Z ] for Z =∑

niZi , ni ∈ Q

but we cannot choose Z with ni ∈ Z

I induction on dimension — this is the original method ofPicard-Poincare-Lefschetz to construct curves on asurface — the picture is

C ′

C

Xs0 Xs

→

J(Xs0) J(Xs)

νZ

where Z = C − C ′, deg Zs =: Z · Xs = 0 andνZ (s) = AJXs (Zs).

Picard and Poincare showed, successfully in my view, thatI any ν = νZ for some Z (Jacobi inversion with

dependence on parameters)I the family J(Xs) has a fixed part of dimension

q = h0(Ω1X )

I varying Z by varying ν in the fixed part traces out ∞q

rationally inequivalent curves on X

Lefschetz showed that the variable part of νZ dependsonly on [Z ], and

(∗)ˆ

[Z ]

ω = 0 for ω ∈ H0(Ω2X );

obvious now but not then — his argument used that

H0(Ω2X ) = dy ∧

(⊕k=2

H0(OP1(k)

)part of R0ωX/P1

)Lefschetz then reversed the argument to show that any νgives a class in [ν] ∈ H2(X ,Z), and that (∗) is satisfiedwith [ν] replacing [Z ]. But then ν = νZ for some Z .

I using mixed Hodge theory this argument generalizes andhas been used to prove a number of non-existence resultsbut (except in a few special cases) no existence ones — itdoes show that associated to a Hodge class ξ there is anadmissible normal function νξ

— recently a number of people have revisited the old ideaof using, instead of just pencils, the whole family S ofhypersurfaces of high degree — the geometric idea is thatwhen dim X = 2n any primitive algebraic n-cycle Z willhave support contained in necessarily singularhypersurfaces Xs — building on earlier work on Neronmodels in higher codimension, using M. Saito’s theoryseveral people have

I shown how to define sing νξ for an admissible normalfunction νξ

I shown that over Q, ξ = [Z ]⇔ sing νξ 6= ∅ for d 0

from examples one may suspect something like

d 0 means d = C |ζ2|

where the constant is uniform in the moduli of X — buteven formulating this precisely seems to run into theprevious issue that even if ζ = [Z ] we don’t see how tobound the denominators in the ni

What is at issue is what one might think of as theeffective Hodge conjecture. If this were done and theabove remark about singνξ reformulated as

Sνξ−→ J = universal family of Neron models

ν−1ξ (Jsing) = sing νξ

then at least the topological question

ν∗ξ [Jsing] 6= 0 for d 0

would make sense.