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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
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
joint w/ Ugo Bruzzo
Int. J. Math , vol. 25, (2), 2014
Donagi–Markman cubic for the generalised Hitchin system
2 / 29
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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