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Donagi–Markman

cubic for thegeneralised

Hitchin system

Peter Dalakov

Intro

CameralCovers

Abelianisationand generalisedPryms

Proof ofTheorem A

Cubiccondition andTheorem B

Intro Cameral Covers Abelianisation and generalised Pryms Proof of Theorem A Cubic condition and Theorem B

Donagi–Markman cubic for the generalised Hitchinsystem

Peter Dalakov

IMI, Sofia, Bulgaria

June 19, 2014

1 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



joint w/ Ugo Bruzzo

Int. J. Math , vol. 25, (2), 2014

Donagi–Markman cubic for the generalised Hitchin system

2 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Goal:

Describe the infinitesimal period map for the generalised (ramified)Hitchin system

h : HiggssmG,D,c → B

Upshot:

The Balduzzi–Pantev formula still holds along (good) symplecticleaves

3 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Data and Notation

We work in the holomorphic category!

Fix the following

1 Geometric dataX smooth, compact, connected RS of genus g ≥ 0D ≥ 0 divisor on X, s.t. KX(D)2 is very ample

2 Lie-theoretic dataG simple complex Lie groupT ⊂ B ⊂ G Cartan and Borel subgroupsR+ ⊂ t∨ positive roots

3 NotationL := KX(D)t ⊂ b ⊂ g the Lie algebras of T ⊂ B ⊂ Gl = rkg = dim t the rank of GW = NG(T )/T the Weyl groupdi degrees of basic G-invariant polynomials on g (i = 1 . . . l)

4 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



To these data one can associate two (closely related) families ofabelian varieties:

1 Certain moduli space of “meromorphic” Higgs bundles on X

2 A family of generalised Pryms for a family of (branched)W -Galois covers of X

The family 1 is the one that appears in the title.

The family 2 is easier to work with.

They have the same (infinitesimal) period map.

5 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Family 1

A holomorphic L-valued G-Higgs bundle on X is a pair (E, θ), where

E → X is a (holomorphic) principal G-bundle

θ ∈ H0(X, adE ⊗ L)

Coarse moduli space of semi-stable Higgs bundles

HiggsG,D =∐

c∈π1(G)

HiggsG,D,c

Proper morphism (Hitchin map)

hc : HiggsG,D,c → B

induced by the adjoint quotient g→ g G ' t/W

Markman (’94,’00), Bottacin (’95):

HiggsG,D,c is holomorphic Poisson

hc is an ACIHS

6 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Poisson structure (after Markman)

B ' H0

(X,

l⊕i=1

Ldi

)

B0 := H0

(X,

l⊕i=1

Ldi(−D)

)⊂ B

PstG,D,c = moduli space of stable framed G-bundles of topologicaltype c with level-D structureGD = level group

T∨PstG,D,cµ

%%LLLLLLLLLLL

xxqqqqqqqqqq

HiggssmG,D,c

hc

hc

&&NNNNNNNNNNNg∨D

B // B/B0

' // g∨D GD

.

7 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Symplectic leaves ! Coadjoint orbits

S = µ−1(O)/GD! O =∏i

Oi ⊂ g∨D

Every hc-fibre contains a unique leaf of maximal rankRestrict to “generic locus” B ⊂ B

To o ∈ B corresponds a smooth “cameral cover” with simpleGalois ramification

tot t⊗C L ⊃ Xoπo //X

An o ∈ B ⊂ B corresponds to regular orbit Oo

h−1c (o+ B0) = S ⊂ HiggsG,D,c

Intersecting with the generic locus:

B := (o+ B0) ∩B ⊂ B

HiggsG,D,c ⊃ S|B = h−1 (B)hB // (B, o)

8 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Theorem A (U.Bruzzo, P.D.)

Let o ∈ B. There exists a natural isomorphism

TB,o ' H0(Xo, t⊗C KXo)W .

The differential at o of the period map for the (maximal rank)symplectic leaf S|B → B is

co : H0(Xo, t⊗C KXo)W −→ Sym2

(H0(Xo, t⊗C KXo

)W)∨

co(ξ)(η, ζ) =1

2

∑p∈Ram(πo)

Res2p

(π∗oLYξ (D)

D

∣∣∣∣o×X

η ∪ ζ

).

Here:

Yξ 7→ ξ under TB,o ' H0(Xo, t⊗C KXo)W

D is the discriminant

L is the Lie derivative.

9 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Plan for the rest of the talk:

1 Cameral covers

2 Generalised Pryms and abelianisation

3 Sketch of proof of Theorem A

4 Cubic condition and Theorem B

10 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Adjoint Quotient

W acts on t by reflections

C[t] = Sym t∨ ⊃ C[t]W

Non-canonically: C[t]W ' C[I1, . . . , Il], deg Ik = dk

C×-equivariant quotient morphism

χ : t→ t/W ' Cl

χ(v) = (I1(v), . . . , Il(v))

Ramified W -cover, branched over a singular hypersurface

Bra(χ) = Z(P ) ⊂ t/W ' Cl,

the zero locus of the discriminant

Dχ =∏α∈R

α = P (I1, . . . , Il) ∈ C[t]W

11 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Twist χ with L to get a W -cover

tot t⊗C Lp // tot t⊗C L/W

tot L⊕l // tot⊕

i Ldi

Hitchin base:

B = H0(X, t⊗C L/W ) '⊕i

H0(X,Ldi).

Evaluation morphisms

b ∈ B evb : X → tot t⊗C L/W

ev : B ×X → tot t⊗C L/W.

12 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Universal cameral cover

Pull back the W -cover p by ev:

Xb //

πb

X ι //

π

tot t⊗C L

p

b ×X // B ×X ev

//

""FFFFFFFFF tot t⊗C L/W

q

yyssssssssss

X

Denote r = q p : tot t⊗C L→ X

In a local trivialisation, X is given by∣∣∣∣∣∣I1(α1, . . . , αl) = b1(β, z)

. . .Il(α1, . . . , αl) = bl(β, z)

,

13 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Genericity

Dχ induces a section D ∈ H0(Xb, r∗L|R|)W

X ι //

π

tot t⊗C L

p

D

''OOOOOOOOOOO

B ×X ev // tot t⊗C L/WD // tot q∗L|R|

.

Generic cameral covers: smooth with simple ramification

evb(X) ∩ Z(D)sing = ∅ evb(X) t Z(D)sm

b ∈ B B.

14 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Generalised Pryms

Fix generic o ∈ B, πo : Xo → XRecall: cochar = Hom(C×,T), cochar ⊗Z C× ' TDonagi–Gaitsgory (’00) introduce two abelian sheaves T ⊃ T on X:

T = πo∗(cochar ⊗O×

Xo

)Wand T :

T (U) :=

t ∈ Γ

(π−1(U), cochar ⊗O×

Xo

)W ∣∣∣∣ α t|Ramα= +1,∀α ∈ R

.

Properties:

T /T is Z/2Z-torsion

If G 6= Bl = SO2l+1, T = TH1(X, T ) and H1(X, T ) are isogenous abelian varieties

Generalised Prym variety: PrymXo/X:= H1(X, T )

15 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Abelianisation Theorem

The work of Donagi–Gaitsgory (’00) implies:

Fibres h−1c (o) are Prym0

Xo/X-torsors

HiggsG,D∣∣B→ B is a torsor for PrymX/B → B

Local sections exist : over U ⊂ B,

HiggsG,D,c∣∣U' Prym0

X/U ,

whenever HiggsG,D,c 6= ∅.

Remarks:

Roughly, we are interested in(cochar ⊗Z PicXo

)W⊂ cochar ⊗Z PicXo

Generalises lots of previous work of Hitchin,Beauville–Narasimhan–Ramanan, Donagi, Scognamillo,Faltings, Beilinson. . .

16 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Step 1: recasting the symplectic structures

For any vector space V , there is a natural t-valued 2-form onV ⊕ (V ∨ ⊗ t):

((x, α⊗ s), (y, β ⊗ t)) = α(y)s− β(x)t

This form induces a meromorphic t-valued 2-form

ωt ∈ H0(tot t⊗C L, t⊗C Ω2t⊗CL(r∗D))W ,

where r : tot t⊗C L→ X.

Have Xo ⊂ tot t⊗C L, so ωt|Xo gives a sheaf homomorphism

N → t⊗C KXo(r∗D)

via0 //TXo // Tt⊗CL|Xo //N //0

In general, this map is not an isomorphism, but induces one oninvariant global sections:

H0(Xo, N(−r∗D))W 'ωt H0(Xo, t⊗C KXo

)W

Idea: [Kjiri,’00] =⇒ HiggsG,D,c satisfies the “rank 2 condition” ofHurtubise–Markman.

17 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Combining with

TB,o ' H0(Xo, N(−r∗D))W

we getTB,o 'ωt H

0(Xo, t⊗C KXo)W .

Finally:

Prym0X/B is symplectic and the fibration

Prym0X/B −→ B

is Lagrangian.Idea: for Po = Prym0

Xo/X, we have

TPo = H1(Xo, t⊗C O)W ⊗OPo

TB,o ' H0(Xo, t⊗C KXo)W .

Under the local identifications S|U ' Prym0X/U the symplectic

structures on both sides coincide.

18 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Step 2: Use a theorem of Griffiths

Proposition

Let πo : Xo → X be the cameral cover, corresponding to a genericpoint o ∈ B. Then the differential of the period map ofhB : S|B → B at o is given by

co : H0(Xo, t⊗C KXo)W −→ Sym2

(H0(Xo, t⊗C KXo

)W)∨

,

co(ξ)(η, ζ) =1

2πi

∫Xo

κ(Yξ) ∪ η ∪ ζ,

where κ : TB,o → H1(Xo, TXo) is the Kodaira–Spencer map of thefamily X|U → U , and Yξ 7→ ξ under the isomorphism

TB,o 'ωt H0(Xo, t⊗C KXo

)W .

19 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Idea of proof:

Consider a proper holomorphic submersion

h : H → B

H, B complex manifolds

Hb = h−1(b): polarised compact connected Kahler manifolds(polarisation varies smoothly with b ∈ B)

This gives rise to a weight-1 polarised Z-VHS on B(F•,FZ,∇GM , S

)and a period map Φ.

20 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Griffiths related dΦ to ∇GM and the Kodaira–Spencer map κ

Theorem (Griffiths, ’68)

The infinitesimal period map satisfies

TB,o

κ$$IIIIIIIIII

dΦo=gr ∇GM // TD,Φ(o)

H1(Ho, T )

m∨

99ssssssssss

gr∇GM : F1 −→ Ω1B ⊗F0/F1 ' Ω1

B ⊗F1∨

Cup product H1(T )×H0(Ω1)→ H1(O) induces

m∨ : H1(T )→ H1(O)⊗H0(Ω1)∨ 'S(F1∨o

)⊗2

21 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



The Proposition follows from Griffiths’ Theorem, using that

H1(Po, TPo) ' H1(Xo, t⊗C O)W⊗2 and polarisation-preservingdeformations are contained in

Sym2H1(Xo, t⊗C O)W ' Sym2H0(Xo, t⊗C KXo)W∨

the polarisation on the Prym is determined by the polarisationon Xo

the map m∨ from Griffiths’ theorem is dual to themultiplication map

m : H0(Xo, t⊗CKXo)W⊗2 → H0(Xo, t⊗ t⊗K2)→tr H

0(X0,K2)

If V is a (f.d.) vector space, the natural isomorphismHom(V ∨,Hom(V ∨, V )) = Hom(V ∨⊗3,C) is given by

F 7−→ (Y ⊗ α⊗ β 7→ β(F (Y )(α))) .

22 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Step 3: Calculate the Kodaira–Spencer map for X|B → BIdea: Use a cover, adapted to the dynamics of ramification data

Xo //

πo

X ι //

π

tot t⊗C L

p

o ×X // B×X ev

// tot t⊗C L/W

Take U ⊂ (B, o), contractibe, and a cover U ∪V = XU , where:

1 U := XU\Ram(π)2 V ⊃ Ram(π), more intricate:

Bra(πo) = p1, . . . , pN, N = |R| degL(Uj , zj)j → X, where U0 = X\p1, . . . , pN, and Uj 3 pjnon-intersecting open disks; assume supp(D) ⊂ U0

Unramified cover U ×X ⊃ Bra(π)|U → U . Take holomorphicsections cj : U → X. Assume that cj(o) = pj , and cj(U) ⊂ UjV := π−1

(∐j 6=0 graph cj

)⊂ X|U

V =∐

Vjα,

Vjα = π−1 (graph cj) ∩ Z(α) =∐k

Vkjα

23 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Global Kodaira–Spencer map:

κ : Γ(TU )→ R1f∗TX/U

Restricting:κ = κ|o : TU,o → H1(Xo, TXo)

By genericity, on Vkjα, α := α1 the ramification locus of π is:∣∣∣∣ α2 = (z − c(β))v(β, z)

αi = gi(α, z), i ≥ 2

For a vector field Y = ∂∂β

,

∂βD

D=∑α∈R+

π∗∂βα

2

α2

καz(Y ) =α

2π∗∂βα

2

α2

∣∣∣∣z=ϕ(β,α)

∂

∂α= − ∂βc

2αu(β, α)

∂

∂α+ . . . ,

and

Resα=0

(καzyg(α)dα2) =

1

2Res2

α=0

(π∗∂βD

Dg(α)dα2

).

24 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Lagrangian structures

Let h : H → B be a family of polarised abelian varieties (complextori).

Then:

We haveV = h∗TH/B ' F1∨,

TH/B = h∗V

If (H, σ) – holomorphic Poisson, σ ∈ Γ(H,Λ2TH

), with

Lagrangian fibres Xb = h−1(b), then get

i =cσ : V∨ = Ω1H/B → TB

with integrable image

25 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Relating σ and i:

The fibration h : H → B induces a filtration

0 //F //Λ2TH //h∗Λ2TB //0

0 //h∗Λ2V //F //h∗ (V ⊗ TB) //0

and

H0(H,Λ2TH) // H0(H, h∗(V ⊗ TB))

σ // h∗(i)

26 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Weak cubic condition, Poisson form

Theorem (Donagi–Markman,’93)

Let h : H → B be a fibration of PAV (complex tori) with period mapΦ.Fix i : V∨ → TB, with integrable image.Then H admits σ ∈ H0(Λ2TH), inducing the given i, and making thefibres Lagrangian ⇐⇒

dΦ i : V∨ −→ Sym2V

comes from a cubic c ∈ Sym3V under

Sym3V → V ⊗ Sym2V

If so, there is unique such σ, making the zero section Lagrangian.

27 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Locally on B, symplectic case(U , o) ⊂ B – contractible open

A choice of marking gives a trivialisation

V ' Vo ⊗C O, Vo = H0(Ω1Ho)∨

and combined with i:

Vo ⊗C OU ' T∨U

With respect to the marking and a basis of Vo, Φ is representedby period matrices (∆δ, Z(u)), u ∈ U .

∆δ = diag(δ1, δ2, . . .) Zt = Z, ImZ > 0

Canonical symplectic form on T∨U ' VU descends toH|U ' T

∨U /F∨Z iff

Z = Hess(F),

F : U → C “holomorphic prepotential”

Then

c =∑ ∂3F

∂ai∂aj∂akdai · daj · dak

in suitable coordiantes28 / 29

Donagi–Markman


Hitchin system

Peter Dalakov

Intro

CameralCovers


Proof ofTheorem A



Theorem B (U.Bruzzo, P.D.)

With the same notation as above,

co(ξ, η, ζ) =∑

p∈Ram(πo)

Res2p

∑α∈R

α(ξ)α(η)α(ζ)

α(λo),

where λo ∈ H0(Xo, t⊗C L) is the tautological section.

29 / 29

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