On Hitchin’s Connection.

Bert van Geemen and Aise Johan de Jong

September 22, 1997

1 Introduction

1.1

The aim of this paper is to give an explicit expression for Hitchin’s connection in the case of rank 2 bundles with trivial determinant over curves of genus 2.

We sketch the general situation. Let π : C → S be a family of projective smooth curves. Let p : M → S be the family of moduli spaces of (S-equivalence classes of) rank r bundles with trivial determinant on C over S. On M we have a naturally defined determinant linebundle L. The vector bundles p_{∗}(L^{⊗k}) on S have a natural (projective) flat connection (such a connection also exists for other structure groups besides SL(r)).

This connection first appeared in Quantum Field Theory (conformal blocks) and was subsequently studied by various mathematicians (see [Hi] and [BM] and the references given there). We follow the approach of Hitchin [Hi] who showed that this connection can be constructed along the lines of Welters’ theory of deformations [W].

In case the curves have genus 2 and the rank r = 2, the family of moduli spaces p : M → S is isomorphic (over S) to a P^{3}-bundle

p : M ≅ P ⟶ S.

Under this isomorphism L corresponds to O_{P} (1) ⊗ p^{∗}N for some line bundle N on S. Giving a projective connection on p_{∗}L^{⊗k} is then the same as giving a projective connection on p_{∗}O_{P} (k)

∇^{(k)} : p_{∗}O_{P} (k) ⟶ p_{∗}O_{P} (k) ⊗ Ω_{S}^{1}.

1.2

In the first part of the paper (Section 2) we give a characterization of Hitchin’s connection (strictly speaking, we define a new connection which is almost characterized by Hitchin’s requirements).

We first develop a bit of general theory concerning heat operators. In the situation above a heat operator is a certain type of map D : p^{∗}(T_{S}) → Diff_{M}^{(2)}(L^{⊗k}). Thus a vector field θ on S gives a second order differential operator D(θ) on L^{⊗k}, which determines a map D(θ) : p_{∗}L^{⊗k} → p_{∗}L^{⊗k}. These then determine a connection on p_{∗}L^{⊗k}. The symbol of D is a map ρ_{D} : p^{∗}(T_{S}) → S^{2}T_{M/S}.

Let M^{s} be the locus of stable bundles in M. The codimension of the non-stable locus in M is at least two if g ≥ 3 so that p_{∗}L^{⊗k}|_{M^{s}} = p_{∗}L^{⊗k} in this case. Hitchin showed for g ≥ 3 that if a heat operator D on L^{⊗k} over M^{s} determines the desired connection on p_{∗}L^{⊗k}, then its symbol map:

ρ_{D}^{s} : p^{∗}(T_{S}) ⟶ S^{2}T_{M^{s}/S},

must be equal to a map 1/(k + 2)ρ_{Hitchin}^{s}. The map ρ_{Hitchin}^{s} is determined by the deformation theory of a typical fibre C and of the stable bundles on C. He also proved that ρ_{Hitchin}^{s} extends to a ρ_{Hitchin} : p^{∗}(T_{S}) → S^{2}T_{M/S} for bundles of rank r on curves of genus g, except maybe for the case r = 2, g = 2 (!).

We show, using simple bundles, that ρ_{Hitchin}^{s} also extends to a ρ_{Hitchin} in the case g = 2 (Theorem 2.6.1). From a study of heat operators and their symbols we conclude that we can find a (projective) heat operator D with the right symbol over all of M for g = 2 (Corollary 2.6.3). Such a D need not be unique however. To pin it down we use the action of the twotorsion group scheme H := P ic_{C/S}^{0}[2] on M (tensoring a rank two bundle by a line bundle of order two doesn’t change the determinant). Then H acts (projectively) on L^{⊗k}, this action can be linearized to the action of a central extension G of H on L^{⊗k}

1 ⟶ G_{m} ⟶ G ⟶ H ⟶ 0.

We show that there is a unique (projective) heat operator

D_{k} : p^{∗}(T_{S}) ⟶ Diff_{M}^{(2)}(L^{⊗k}) with ρ_{D_{k}} = \frac{1}{k+2}ρ_{Hitchin}

which commutes with the action of G (Proposition 2.6.4). The (projective) connection determined by D_{k} is called Hitchin’s connection (cf. 2.6.6).

1.3

To be able to compute D_{k} (and to establish some of the results mentioned above), we use a map f : P ic_{C/S}^{0} → M over S whose image is the family of Kummer surfaces K → S. We show that the pull-back of D_{k} along f determines the connection on q_{∗}f^{∗}L^{⊗k} given by the heat equations on (abelian) theta functions. (The connection defined by D_{k} on p_{∗}L^{⊗k} does not pull-back to the ‘abelian’ connection however.)

We show that D_{k} is characterized by this property. Moreover, we express this property in terms of the (local) equation defining K ⊂ M (2.6.7 (Eq.)).

1.4

To write down a completely explicit connection (on p_{∗}O_{P} (k)) we must now choose a certain S and a family of curves C over S. We take S = P, the configuration space of 2g + 2 points in C^{2g+2} and take the obvious family of hyperelliptic curves over it:

C  ⊃ C_{z} : y^{2} = ∏_{i}(x − z_{i})
↓  ↓
P := C^{2g+2} − {(z_{1}, …, z_{2g+2}) : z_{i} = z_{j} if i ≠ j }, ∋ z

In the case g = 2 this family dominates the moduli space of genmus 2 curves (the induced morphism of P to the stack of genus two curves is smooth and surjective). We will define connections on bundles over P for any g. In the case g = 2 this connection is basically the Hitchin connection (see Theorem 4.4.2). It seems possible that for general g the connection is related to the Hitchin connection for the family of hyperelliptic curves above.

For g = 2 we can see which bundle to use: The pull-back of the bundle p : P → P along an unramified 2^{4} : 1-Galois cover P̃ → P with group H = (Z/2Z)^{4} is trivial. The bundle P can thus obtained as a quotient of P̃ × P^{3} by H (4.4.1). Therefore, locally (in the complex or the etale topology), the vector bundles p_{∗}O_{P} (k) and the trivial bundle S_{k} ⊗ O_{P} are canonically isomorphic, with S_{k} = H^{0}(P^{3}, O(k)), the vector space of homogenous polynomials of degree k in 4 variables. Thus it suffices to write down the connection on the trivial bundle which corresponds to Hitchin’s connection on p_{∗}O_{P} (k).

This connection on S_{k} ⊗ O_{P} (for general g) is given by

∇ : S_{k} ⊗ O_{P} ⟶ S_{k} ⊗ Ω_{P}, ∇(w ⊗ f) = w ⊗ df − \frac{1}{k+2} ∑_{i≠j} M_{ij}(w) ⊗ \frac{fdz_{i}}{z_{i} − z_{j}} ,

for certain endomorphism M_{ij} ∈ End(S_{k}). We give two descriptions of the endomorphisms M_{ij}. One is based on the Lie algebra so(2g + 2) and its (half)-spin representations. We will in fact identify S_{k} = S^{k}V (ω_{g+1}) where V (ω_{g+1}) is a half spin representation of so(6).

The other description uses the Heisenberg group action and we sketch that description here. The family C has 2g + 2 sections which are the Weierstrass points on each fiber:

P_{i} : P ⟶ C, z ⟼ (x, y) = (z_{i}, 0).

Each P_{i} − P_{j} is a section in P ic_{C/S}^{0}[2](S) ≅ (Z/2Z)^{2g}. There is a central extension G of this group which acts on the S_{k}’s (the groupscheme G is a twist of the constant group scheme G, these group schemes are isomorphic over P̃). Let ±U_{ij} ∈ End(S_{1}) be the endomorphisms induced by P_{i} − P_{j} with the property that U_{ij}^{2} is the identity. Since End(S_{1}) = S_{1} ⊗ S_{1}^{∗} and S_{1}^{∗} may be identified with the derivations on S := ⊕S_{k} and the composition U_{ij}U_{ij} may be considered as a second order differential operator which acts on any S_{k}. The symbol (that is, the degree two part) of this operator is, up to a factor −1/16, the desired M_{ij} (which, in this sense, does not depend on k).

The idea for trying these Ω_{ij}’s comes from [vGP]. There the Hitchin map, a Hamiltonian system on the cotangent bundle T^{∗}M^{s}, is studied. The Hitchin map is basically the symbol of the heat operator D_{k} (cf. [Hi] §4). Among the results of [vGP] is an explicit description of the Hitchin map for g = 2 curves (obtained from line geometry in P^{3}) and implicitly in that description is the one for the M_{ij} given here.

1.5

Now that the connection is determined it is interesting to consider its monondromy representation. In the case of rank two bundes, this has been considered by Kohno [K2]. We verify that our connection gives has the same local monodromy as Kohno’s representation in Subsection 4.5. Here local monodromy means “at ∞”, e.g., we determine the eigenvalues of the monodromy operator of a small loop around the divisor z_{1} = z_{2} in P. Kohno (and the physicists) use (locally on the base it seems) the Knizhnik-Zamolodzhikov equations (which define a flat connection over P using the Lie algebra sl(2) [Ka], Chapter XIX). It is not clear to us how the KZ-equations are related to our connection (which is based on so(2g + 2)). We do not know which representations of the braid group one obtains from our connections.

2 Characterizing Hitchin’s connection

2.1 Introduction

We recall some deformation theory as described by Welters [W] and relate it to connections. Then we introduce heat operators and show how they define (projective) connections on certain bundles. Next we give a criterion for the existence of a heat equation with a given symbol and in Section 2.4 we recall the definition of the symbol of Hitchin’s connection. In Section 2.5 we combine these results and give criteria (in 2.6.7 and 2.6.6) which determine Hitchin’s connection.

2.2 Deformation theory

2.2.1 Deformations

Let X be a smooth projective variety over C with tangent bundle T_{X}. Let L be an invertible O_{X}-module on X and let s ∈ H^{0}(X, L). We are going to classify first order deformations of X, of the pair (X, L) and of the triple (X, L, s). See [W] for proofs.

Isomorphism classes of first order deformations X_{ϵ} of X (over Spec C[ϵ]/(ϵ^{2})) are classified by H^{1}(X, T_{X}). We write [X_{ϵ}] for the class in H^{1}(X, T_{X}) of the deformation X_{ϵ}; similar notation will be used throughout.

The first order deformations (X_{ϵ}, L_{ϵ}) of the pair (X, L) are classified by H^{1}(X, Diff_{X}^{(1)}(L)). Here Diff_{X}^{(1)}(L) is the sheaf of first order differential operators on L. This sheaf sits in an exact sequence:

0 ⟶ O_{X} ⟶ Diff_{X}^{(1)}(L) ⟶^{σ} T_{X} ⟶ 0

where the map σ : Diff_{X}^{(1)}(L) → T_{X} gives the symbol of the operator. This sequence gives a map α : H^{1}(X, Diff_{X}^{(1)}(L)) → H^{1}(X, T_{X}), which maps [(X_{ϵ}, L_{ϵ})] to [X_{ϵ}].

Finally, we consider first order deformations of the triple (X, L, s). The evaluation map d^{1}s : Diff_{X}^{(1)}(L) → L, D ↦ Ds gives a complex:

0 ⟶ Diff_{X}^{(1)}(L) ⟶^{d^{1}s} L ⟶ 0.

Let H^{1}(d^{1}s) be the first hypercohomology group of this complex. This is the space classifying the isomorphism classes of deformations (X_{ϵ}, L_{ϵ}, s_{ϵ}) of (X, L, s) ([W], Prop. 1.2). The spectral sequence connecting hypercohomology to cohomology gives a map β : H^{1}(d^{1}s) → H^{1}(Diff_{X}^{(1)}(L)), which maps [(X_{ϵ}, L_{ϵ}, s_{ϵ})] to [(X_{ϵ}, L_{ϵ})].

Any element of H^{1}(d^{1}s) may be represented by a Cech cocycle. Let us choose an affine open covering U : X = ⋃ U_{i} and write U_{ij} = U_{i} ∩ U_{j}. A Cech cocycle is given by a pair ({t_{i}}, {D_{ij}}) satisfying the relations:

t_{i} − t_{j} = D_{ij}(s), D_{jk} − D_{ik} + D_{ij} = 0 (t_{i} ∈ L(U_{i}), D_{ij} ∈ Diff_{X}^{(1)}(L)(U_{ij})).

The maps β and α ∘ β correspond to forgetting first s_{ϵ} and then L_{ϵ}. They can be described as follows. The cocycle ({t_{i}}, {D_{ij}}) is mapped to the 1-cocycle {D_{ij}} in the sheaf Diff_{X}^{(1)}(L) under β. This is then mapped to the 1-cocycle {σ^{(1)}(D_{ij})} in the sheaf T_{X}.

Let us make explicit a deformation (X_{ϵ}, L_{ϵ}, s_{ϵ}) associated to ({t_{i}}, {D_{ij}}). Write U_{i} = Spec(A_{i}) and U_{ij} = Spec(A_{ij}). Let M_{i} = L(U_{i}) and M_{ij} = L(U_{ij}). The section s gives elements s_{i} ∈ M_{i}; note that t_{i} ∈ M_{i} as well. Write θ_{ij} = σ(D_{ij}) ∈ T_{X}(U_{ij}). The 1-cocycle {θ_{ij}} defines a scheme X_{ϵ} over Spec(C[ϵ]) by glueing the rings A_{i}[ϵ] via the isomorphisms

A_{ij}[ϵ] ⟶ A_{ij}[ϵ], f + gϵ ⟼ f + (θ_{ij}(f) + g)ϵ.

Here we view θ_{ij} as a derivation. The 1-cocycle {D_{ij}} in Diff_{X}^{(1)}(L) gives a deformation L_{ϵ} of L by glueing the A_{i}[ϵ]-modules M_{i}[ϵ] via the A_{ij}[ϵ]-module isomorphisms:

ϕ_{ij} : M_{ij}[ϵ] ⟶ M_{ij}[ϵ], m_{ij} + m′_{ij}ϵ ⟼ m_{ij} + (D_{ij}(m_{ij}) + m′_{ij})ϵ.

Note one has D_{ij}(fs) = fD_{ij}(s) + θ_{ij}(f)s by definition of the symbol σ and σ(D_{ij}) = θ_{ij}. The cocycle ({t_{i}}, {D_{ij}}) in H^{1}(d^{1}s) defines a deformation s_{ϵ} ∈ H^{0}(X_{ϵ}, L_{ϵ}). The section s_{ϵ} is given by the family of elements:

s_{ϵ} := {s_{ϵ,i}} (∈ M_{i}[ϵ]) with s_{ϵ,i} := s_{i} + t_{i}ϵ.

The cocycle relation t_{i} − t_{j} = D_{ij}(s) shows that the s_{ϵ,i}’s glue: ϕ_{ij}(s_{ϵ,j}) = s_{ϵ,i}.

2.2.2 Second order operators

The following construction is useful to obtain elements in H^{1}(d^{1}s) ([W], (1.9)). Let Diff_{X}^{(2)}(L) be the sheaf of second order differential operators on L. The symbol gives an exact sequence which is the first row of the complex:

0 ⟶ Diff_{X}^{(1)}(L) ⟶ Diff_{X}^{(2)}(L) ⟶ S^{2}T_{X} ⟶ 0
↓  ↓  ↓
0 ⟶  L  ⟶  L  ⟶ 0 ⟶ 0,

the vertical maps are evaluation on s (D ↦ Ds), thus the first column is the complex d^{1}s considered before. From the exact sequence of hypercohomology one obtains a map

δ_{s} : H^{0}(S^{2}T_{X}) ⟶ H^{1}(d^{1}s).

Let us give a description in terms of cocycles of this map. Let U = {U_{i}} be an affine open cover of X as before. Let w ∈ H^{0}(X, S^{2}T_{X}). We can find operators D_{i}^{(2)} ∈ Diff_{X}^{(2)}(L)(U_{i}) which map to w|_{U_{i}} in S^{2}T_{X}(U_{i}). Then

δ_{s}(w) = ({D_{i}^{(2)}(s)}, {D_{i}^{(2)} − D_{j}^{(2)}}) (∈ H^{1}(d^{1}s)).

It is easy to verify that δ_{s}(w) is indeed a cocycle for the complex defined by d^{1}s. Note w = σ(D_{i}^{(2)}) = σ(D_{j}^{(2)}) on U_{ij}, thus D_{i}^{(2)} − D_{j}^{(2)} is a first order operator.

Note that β(δ_{s}(w)) = {D_{i}^{(2)} − D_{j}^{(2)}} (∈ Diff_{X}^{(1)}(L)) is independent of the choice of the section s. Therefore, given w (∈ H^{0}(X, S^{2}T_{X})) we get a deformation (X_{ϵ}, L_{ϵ}) such that any section s ∈ H^{0}(X, L) is deformed to a section s_{ϵ} ∈ H^{0}(X_{ϵ}, L_{ϵ}).

2.2.3 Infinitesimal connections

Let (X_{ϵ}, L_{ϵ}) be a deformation of (X, L). We consider a differential operator D of order at most two on L_{ϵ} which locally can be written as (notations as above, cf. 2.3.2 below)

D = {D_{i}} (∈ H^{0}(X_{ϵ}, Diff_{X_{ϵ}}^{(2)}(L_{ϵ}))), D_{i} = ∂_{ϵ} + R_{i} : M_{i}[ϵ] ⟶ M_{i}[ϵ]

with ∂_{ϵ}(n + ϵm) = m for n, m ∈ M_{i}, and R_{i} a C[ϵ]-linear map of order at most 2 on M_{i}. Thus R_{i} is a second order differential operator involving only deriviatives in the fiber directions (and not in the base direction ϵ). The symbol of such a D is the second order part of the D_{i} (and thus of the R_{i}). The restriction of the symbol to X (⊂ X_{ϵ}) is:

ρ̄(D) := {σ(R̄_{i})} (∈ H^{0}(X, S^{2}T_{X})), where R_{i} = R̄_{i} + ϵR̄′_{i}

with R̄_{i}, R̄′_{i} : M_{i} → M_{i}[ϵ] (recall R_{i} is ϵ-linear).

For any s ∈ H^{0}(X, L) we then have the hyper cohomology class δ_{s}(ρ̄(D)) = ({R̄_{i}(s_{i})}, {R̄_{i} − R̄_{j}}) (∈ H^{1}(d^{1}s)). Let us show that the deformation (X_{ϵ}, L_{ϵ}) is the deformation determined by −β(δ_{s}(ρ̄(D))):

[(X_{ϵ}, L_{ϵ})] + β(δ_{s}(ρ̄(D))) = 0 (∈ H^{1}(X, Diff_{X}^{(1)}(L))).

This can be verified as follows. Note that βδ_{s}(ρ̄(D)) = {R̄_{i} − R̄_{j}} and [(X_{ϵ}, L_{ϵ})] = {D_{ij}}. The fact that D is an operator on L_{ϵ} implies that D_{i} ∘ (id + ϵD_{ij}) = (id + ϵD_{ij}) ∘ D_{j}. Taking the “constant” term of this equation we obtain R̄_{i} + D_{ij} = R̄_{j} as desired.

Next we consider any deformation (X_{ϵ}, L_{ϵ}, s_{ϵ}) of (X, L, s). We say that a section s_{ϵ} = {s_{i} + ϵt_{i}} ∈ H^{0}(X_{ϵ}, L_{ϵ}) is flat for D if

D_{i}(s_{i} + ϵt_{i}) ≡ 0 mod (ϵ), equivalently t_{i} = −R̄_{i}(s_{i}),

in that case we write ∇(D)(s_{ϵ}) = 0. Since [(X_{ϵ}, L_{ϵ}, s_{ϵ})] = ({t_{i}}, {D_{ij}}) (∈ H(d^{1}s)) we get:

δ_{s}(ρ̄(D)) = −[(X_{ϵ}, L_{ϵ}, s_{ϵ})] iff ∇(D)(s_{ϵ}) = 0.

2.2.4 A remark on symbols

We use the conventions of [W] concerning differential operators and symbols. In particular when we write S^{2}T_{X} we mean symmetric tensors in the sheaf of vector fields on X. Note that if X = A_{C}^{1} with coordinate t, then the symbol of the operator ∂^{2}/∂t^{2} (∈ Diff_{O}^{(1)}) is the symmetric section 2∂/∂t ⊗ ∂/∂t (∈ S^{2}T_{X}). By abuse of notation we will sometimes write σ(∂^{2}/∂t^{2}) = 2∂^{2}/∂t^{2}.

2.3 Heat operators

2.3.1 Notation

Let p : X → S be a smooth surjective morphism of smooth varieties over C. Let L be an invertible O_{X} -module over X.

We write T_{X} (resp. T_{S}, resp. T_{X /S}) for the sheaf of vector fields on X (resp. S, resp. X over S). We denote Diff_{X}^{(k)}(L) the sheaf of differential operators of order at most k on L over X. Further, Diff_{X /S}^{(k)}(L) ⊂ Diff_{X}^{(k)}(L) denotes the subsheaf of p^{−1}(O_{S})-linear operators. We remark that Diff_{X}^{(k)}(L) is a coherent (left) O_{X} -module and that Diff_{X /S}^{(k)}(L) is a coherent submodule. For any k there is a symbol map

σ^{(k)} : Diff_{X}^{(k)}(L) ⟶ S^{k}T_{X} = Sym_{O_{X}}^{k} (T_{X}).

It maps Diff_{X /S}^{(k)}(L) into S^{k}T_{X /S}.

Consider a point x ∈ X (C), an (étale or analytic) neighbourhood U of p(x) in S, and an (étale or analytic) neighbourhood V of x in X such that p(V) ⊂ U. Assume we have coordinate functions t_{1}, …, t_{r} ∈ O_{S}(U) and functions x_{1}, …, x_{n} ∈ O_{X} (V) such that t_{1} = t_{1} ∘ p, …, t_{r} = t_{r} ∘ p, x_{1}, …, x_{n} are coordinates on V. (In the étale case this means that Ω_{U}^{1} = ⨁ O_{U}dt_{i} and Ω_{V/U}^{1} = ⨁ O_{V} dx_{i}.) Then the elements ∂/∂x_{i} form a basis for T_{X /S}|_{V} over O_{V}. The system (V, t_{1}, …, t_{r}, x_{1}, …, x_{n}) will be called a coordinate patch. We can find coordinate patches around any point x ∈ X (C).

2.3.2 Definition

A heat operator D on L over S is a (left) O_{X} -module homomorphism

D : p^{∗}(T_{S}) ⟶ Diff_{X}^{(2)}(L)

which satisfies the following property: for any coordinate patch (V, t_{1}, …, t_{r}, x_{1}, …, x_{n}) as above, and any trivialization L|_{V} ≅ O_{V}, the operator D(∂/∂t_{1}) has the following shape:

D(\frac{∂}{∂t_{1}} ) = f + \frac{∂}{∂t_{1}} + ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} + ∑_{1≤i≤j≤n} f_{ij} \frac{∂^{2}}{∂x_{i}∂x_{j}}

for certain f, f_{i}, f_{ij} ∈ O_{X} (V). More precisely, let

W_{X /S}(L) = Diff_{X}^{(1)}(L) + Diff_{X /S}^{(2)}(L) ⊂ Diff_{X}^{(2)}(L)

be the subsheaf of second order differential operators on L whose symbol lies in S^{2}(T_{X /S}) ⊂ S^{2}(T_{X}). Note that Diff_{X /S}^{(1)}(L) ⊂ W_{X /S}(L) and that there is a canonical exact sequence

0 ⟶ Diff_{X /S}^{(1)}(L) ⟶ W_{X /S}(L) ⟶ p^{∗}(T_{S}) ⊕ S^{2}T_{X /S} ⟶ 0.

Our condition on D above means that D is a map D : p^{∗}(T_{S}) → W_{X /S}(L) whose composition with the map W_{X /S}(L) → p^{∗}(T_{S}) is the identity. (So D(∂/∂t_{1}) has no terms involving ∂/∂t_{i} except for the term ∂/∂t_{1} with coefficient 1.)

Such a map D is determined by the O_{S}-linear map p_{∗}D : T_{S} → p_{∗}(Diff_{X}^{(2)}(L)). Note that O_{S} is a subsheaf of p_{∗}(Diff_{X}^{(2)}(L)). Any O_{S}-linear map D̄ : T_{S} → p_{∗}(Diff_{X}^{(2)}(L))/O_{S} can locally be lifted to a map into p_{∗}(Diff_{X}^{(2)}(L)). This means that any point s ∈ S(C) has a neighbourhood U such that there is an O_{U}-linear map D_{U} : T_{S}|_{U} → p_{∗}(Diff_{X}^{(2)}(L))|_{U} which reduces to D̄|_{U}.

A projective heat operator D̄ on L over S is an O_{S}-linear map

D̄ : T_{S} ⟶ p_{∗}(Diff_{X}^{(2)}(L))/O_{S}

such that any local lifting D_{U} as above gives rise to a heat operator on L over U. The symbol of a heat operator D is the O_{X} -linear map

ρ_{D} : p^{∗}(T_{S}) ⟶ S^{2}T_{X /S}

given by composing D with the symbol σ^{(2)}. We note that the symbol of a projective heat operator is well-defined.

2.3.3 Heat operators and connections

We claim that a heat operator D gives rise to a connection

∇(D) : p_{∗}(L) ⟶ p_{∗}(L) ⊗_{O_{S}} Ω_{S}^{1}.

Indeed, suppose that s ∈ p_{∗}(L)(U) = L(p^{−1}(U)) and that θ ∈ T_{S}(U). Then D(p^{−1}θ) is a second order differential operator on L over p^{−1}(U). Hence D(p^{−1}θ)(s) ∈ L(p^{−1}(U)) = p_{∗}(L)(U). By our local description of D above we see that D(p^{−1}θ)(fs) = fD(p^{−1}θ)(s) + θ(f)s for every f ∈ O_{S}(U). Hence the operation (θ, s) ↦ D(p^{−1}θ)(s) defines a connection ∇(D) on p_{∗}L over S. In the sequel we will often write D(θ) in stead of D(p^{−1}θ).

If we have a projective heat operator D̄, the connection ∇(D̄) = ∇(D_{U}) is only defined locally, by choosing lifts D_{U}. The difference of two local lifts is given by a local section η ∈ Ω_{S}^{1}(U), in which case the difference of the two connections ∇(D_{U}) is multiplication by η. Thus D̄ defines unambiguously a projective connection ¯∇(D̄).

2.3.4 Heat operators and deformations

Let D be a heat operator on L over S. Let 0 ∈ S(C) be a point of the base. Put X = X_{0} and L = L_{0}. An element θ ∈ T_{0}S will also be considered as a morphism θ : Spec(C[ϵ]) → S.

By base change we get a pair (X_{ϵ}, L_{ϵ}) over Spec(C[ϵ]). Clearly, w_{θ} := ρ_{D}(θ) is an element of H^{0}(X, S^{2}T_{X}). It can be seen by working through the definitions that [(X_{ϵ}, L_{ϵ})] + β(δ_{s}(w_{θ})) = 0 for any section s ∈ H^{0}(X, L). (Note that in Subsection 2.2.2 we showed that β(δ_{s}(w_{θ})) was independent of s.) Let s_{ϵ} ∈ H^{0}(X_{ϵ}, L_{ϵ}) lift s. One can show that [(X_{ϵ}, L_{ϵ}, s_{ϵ})] = −δ_{s}(w_{θ}) if and only if the section s_{ϵ} is horizontal for the connection ∇(D).

2.3.5 Projective heat operators and change of line bundle

Let D be a heat operator on L over S. Let g ∈ H^{0}(S, O_{S}^{∗}) be an invertible function on S. Multiplication by p^{−1}(g) defines an invertible operator on L, also denoted by g. For any local section θ of T_{S}, we have the obvious relation

g^{−1} ∘ D(θ) ∘ g = g^{−1}θ(g) + D(θ).

We conclude that g^{−1} ∘ D ∘ g is a heat operator and that the projective heat operators D̄ and \overline{g^{−1} ∘ D ∘ g} are equal. (The difference is given by the section g^{−1}dg = d log g of Ω_{S}^{1}.)

Suppose that M is an invertible O_{S}-module on S. The above implies that the set of projective heat operators on L over S can be identified canonically with the set of projective heat operators on L′ = L ⊗_{O_{X}} p^{∗}(M). Indeed, choose a covering of S on whose members U_{i} the line bundle M becomes trivial. This identifies the heat operators on L and L′ over U. The difference in the local identifications is given by the 1-forms d log g_{ij} ∈ Ω_{U_{ij}}^{1}.

2.3.6 Heat operators and flatness

We say that a heat operator D on L over S is flat if D([θ, θ′]) = [D(θ), D(θ′)] for any two local sections θ, θ′ of T_{S} on U ⊂ S. It suffices to consider local vector fields θ, θ′ on S with [θ, θ′] = 0 and to check that [D(θ), D(θ′)] = 0. We remark that in this case the operator [D(θ), D(θ′)] is a section of Diff_{X /S}^{(3)}(L) over p^{−1}(U). We say that D is projectively flat if we have D([θ, θ′]) = h + [D(θ), D(θ′)] for some function h = h_{θ,θ′} ∈ O_{S}(U) for any θ, θ′ ∈ T_{S}(U). A projective heat operator D̄ is called projectively flat if any of the local lifts D_{U} are projectively flat.

We remark that if the heat operator D is flat, then the associated connection ∇(D) on p_{∗}L is flat as well. In particular, if p_{∗}L is a coherent O_{S}-module, then this implies that p_{∗}L is locally free. In the same vein we have that a projectively flat projective heat operator D̄ defines a projectively flat projective connection ¯∇(D̄) on p_{∗}L.

2.3.7 Heat operators with given symbol

Let X → S and L be as above, and assume that we are given a O_{X} -linear map ρ : p^{∗}T_{S} → S^{2}T_{X /S}. The question we are going to study is the following:

When can we find a (projective) heat operator D on L over S, D : p^{∗}T_{S} → W_{X /S} (⊂ Diff_{X}^{(2)}(L)) with ρ_{D} = ρ?

So the symbol ρ_{D} (which is the composition of D with the map σ^{(2)} : Diff_{X}^{(2)}(L)) → S^{2}T_{X /S}) should be equal to the given ρ.

Let us define a number of canonical maps associated to the situation. First we have the standard exact sequence

0 ⟶ T_{X /S} ⟶ T_{X} ⟶ p^{∗}T_{S} ⟶ 0.

This gives us a O_{S}-linear map (the Kodaira-Spencer map)

κ_{X /S} : T_{S} ⟶ R^{1}p_{∗}T_{X /S}.

Next we have the exact sequence

0 ⟶ T_{X /S} ⟶ Diff_{X /S}^{(2)}(L)/O_{X} ⟶ S^{2}T_{X /S} ⟶ 0.

This gives an O_{S}-linear map

μ_{L} : p_{∗}S^{2}T_{X /S} ⟶ R^{1}p_{∗}T_{X /S}.

The invertible O_{X} -module L is given by an element of H^{1}(X , O_{X}^{∗}). Using the map d log : O_{X}^{∗} → Ω_{X /S}^{1} we get an element of H^{1}(X , Ω_{X /S}^{1}), and hence a section [L] in H^{0}(S, R^{1}p_{∗}(Ω_{X /S}^{1})). This will be called the cohomology class of L. By cupproduct with [L] we get another map p_{∗}S^{2}T_{X /S} → R^{1}p_{∗}T_{X /S}, w ↦ w ∪ [L]. (There is a natural pairing S^{2}T_{X /S} ⊗ Ω_{X /S}^{1} → T_{X /S}.) It is shown in [W, 1.16] that we have μ_{L}(w) = −w ∪ [L] + μ_{O}(w).

Now let θ be a local vector field on S, i.e., θ ∈ T_{S}(U), with U affine. We want to find a D(θ) ∈ W_{X /S}(L) with symbol ρ(θ). Recall that W_{X /S}(L) is an extension:

0 ⟶ Diff_{X /S}^{(1)}(L) ⟶ W_{X /S}(L) ⟶ p^{∗}(T_{S}) ⊕ S^{2}T_{X /S} ⟶ 0.

The obstruction against finding a lift D(θ) (∈ W_{X /S}(L)) of the section p^{−1}(θ) ⊕ ρ(p^{−1}θ) of p^{∗}T_{S} ⊕ S^{2}T_{X /S} over p^{−1}(U) is an element o(θ, L) in H^{1}(p^{−1}(U), Diff_{X /S}^{(1)}(L)) = H^{0}(U, R^{1}p_{∗}(Diff_{X /S}^{(1)}(L))) (recall that U was assumed affine). There is an exact sequence

0 ⟶ O_{X} ⟶ Diff_{X /S}^{(1)}(L) ⟶ T_{X /S} ⟶ 0.

We remark that the image of the class o(θ, L) in R^{1}p_{∗}T_{X /S}(U) is the section κ_{X /S}(θ)+μ_{L}(ρ(θ)). The first condition that has to be satisfied for D̄ to exist is therefore:

(∗)  κ_{X /S} + μ_{L} ∘ ρ = 0.

If this condition is satisfied, then we can find a lift of p^{−1}(θ) ⊕ ρ(p^{−1}θ) to an element \overline{D}_{θ} in the sheaf W_{X /S}(L)/O_{X}. This element is well defined up to addition of a section of T_{X /S}. Hence we get a second obstruction in

(∗∗)  Coker(p_{∗}T_{X /S}⟶^{[L]∪}R^{1}p_{∗}O_{X})

Let us assume for the moment that both obstructions vanish for any θ as above. Then we can locally find a D(θ) lifting the element p^{−1}(θ) ⊕ ρ(p^{−1}θ). Thus, if we have a basis θ_{1}, …, θ_{r} of T_{S} over U, then we can define D_{U} by the formula D_{U}(∑ a_{i}θ_{i}) = ∑ a_{i}D(θ_{i}) for certain choices of the elements D(θ_{i}). If there is some way of choosing the elements D(θ) uniquely up to an element of p^{−1}O_{S}(U), e.g. if the map in (∗∗) is injective and p_{∗}(O_{X}) = O_{S}, then D_{U} determines a unique projective heat operator over U. In this case the map ρ determines a unique projective heat operator on L over S.

Here is another set of hypotheses which imply the existence of heat operators with given symbol. Let 0 ∈ S(C) be a point. Put X = X_{0} and L = L_{0} as in 2.3.4. Let θ ∈ T_{0}S and consider (X_{ϵ}, L_{ϵ}) over Spec(C[ϵ]) as in 2.3.4. We know from 2.3.4 that [(X_{ϵ}, L_{ϵ})] + β(δ_{s}(ρ(θ))) = 0 if a heat operator exists. Suppose that

1. R^{1}p_{∗}O_{X} and R^{1}p_{∗}T_{X /S} are locally free on S, and have fibres H^{1}(X, O_{X}) and H^{1}(X, T_{X}) at any point 0 in S(C).

2. for any 0 and θ as above, we have [(X_{ϵ}, L_{ϵ})] + β(δ_{s}(ρ(θ))) = 0.

If these conditions are satisfied, then we can at least find local liftings D_{U} of ρ to a heat operator. Again, we will get a “canonical” projective heat operator if there is a “preferred” way of choosing the local lifts D(θ_{i}).

2.3.8 Heat operators on abelian schemes

In this subsection, we let X → S be an abelian scheme with zero section o : S → X. We assume the line bundle L is relatively ample on X over S, i.e., L defines a polarization of X over S. We will show that there exists a unique projective heat operator on L over S, see [W].

We first remark that in this case the map (∗∗) of 2.3.7 is an isomorphism. This implies that Γ(p^{−1}(U), Diff_{X /S}^{(1)}(L)) = O_{S}(U) for any U ⊂ S. Hence, by the discussions in 2.3.7 we get a unique heat operator as soon as we have a (symbol) map ρ satisfying (∗∗).

We have μ_{O} = 0 in this situation, as we can let elements of S^{2}T_{X /S} act by translation invariant operators (see [W, 1.20]). Hence, μ_{L}(−) = −[L] ∪ (−). Now we use that

p_{∗}(⊗^{2}T_{X /S}) ⟶^{[L]∪} R^{1}p_{∗}T_{X /S} ≅ o^{∗}T_{X /S} ⊗ R^{1}p_{∗}O_{X}

is an isomorphism. Consider ρ := ([L]∪)^{−1} ∘ κ_{X /S}. It is well known that this is a map into p_{∗}S^{2}T_{X /S} and it solves (∗) by definition; it is of course the unique solution to (∗). The unique projective heat operator D̄ = D̄_{L} with ρ = ρD̄ is called the projective heat operator associated to (X /S, L).

We make the remark that if L′ = L^{⊗n}, then we have ρD̄′ = (1/n)ρD̄. It is known that these projective heat operators are projectively flat. This can be seen in a number of ways; one way is to show that p_{∗}(Diff_{X /S}^{(3)}(L)) = O_{S} in this case, compare 2.3.6.

Uniqueness of the construction of D̄ implies that D̄ commutes (projectively) with the action of the theta-group G(L). In particular the projective monodromy group of the (projectively flat) connection ¯∇(D̄) on (p_{∗}L)_{0} normalizes the action of G(L).

Recall that the classical theta functions satisfy a heat equation. This is exactly the heat equation above where the base is the Siegel upper half space.

2.3.9 Heat operators and functoriality

Let X → S and L as above, and let D be a heat operator on L over S. Note that for any morphism of smooth schemes φ : S′ → S there is a pullback heat operator D′ = φ^{∗}(D) on the pull back L′ on X′ = S′ ×_{S} X. We leave it to the reader to give the precise definition. This pullback preserves flatness. If φ^{∗}(p_{∗}L) ≅ p′_{∗}L′, then the connection ∇(D′) is the pullback of the connection ∇(D). In addition pullback is well defined for projective heat operators and preserves projective flatness.

If ψ : U → X is an étale morphism of schemes over S, then the heat operator D induces a heat operator ψ^{∗}(D) on ψ^{∗}(L) over S.

2.3.10 Heat operators and compatibility

Suppose we have a second smooth surjective morphism q : Y → S and that we have a morphism f : Y → X over S. We will assume that f is a submersion. This means that for any point y ∈ Y(C) there is a coordinate patch (V, t_{1}, …, t_{r}, x_{1}, …, x_{c}, y_{1}, …, y_{m}) of f(y) in X such that (f^{−1}(V), t_{1}, …, t_{r}, y_{1} = y_{1} ∘ f, …, y_{m} = y_{m} ∘ f) is a coordinate patch for y on Y and x_{i} ∘ f = 0.

Let L be a line bundle on X as before. Let us say that a (projective) heat operator D on L over S is (weakly) compatible with f : Y → X if whenever we have a coordinate patch (V, t_{1}, …, t_{r}, x_{1}, …, x_{c}, y_{1}, …, y_{m}) as above, and a trivialization L|_{V} ≅ O_{V}, then the operator D(∂/∂t_{1}) has the following symbol:

ρ_{D}(\frac{∂}{∂t_{1}} ) = ∑_{i≤j=1}^{m} f_{ij} \frac{∂^{2}}{∂y_{i}∂y_{j}} + ∑_{i=1}^{c} x_{i} Ξ_{i}

for certain f_{ij} ∈ O_{X} (V) and Ξ_{i} ∈ S^{2}T_{X /S}(V). More precisely, this means there exists a ρ = ρ_{f,D} : q^{∗}T_{S} → S^{2}T_{Y/S} such that the following diagram commutes:

f^{∗}(ρ_{D}) : f^{∗}p^{∗}T_{S} ⟶ f^{∗}S^{2}T_{X /S}
||  ↑
ρ_{f,D} : q^{∗}T_{S} ⟶ S^{2}T_{Y/S}

Suppose that D′ is a heat operator on f^{∗}L on Y over S. We say that D′ is (weakly) compatible with D if D is (weakly) compatible with f and we have ρ_{D′} = ρ_{f,D}. In this case it is not true in general that D(θ)(s)|_{Y} = D′(θ)(s|_{Y}).

Assume that D is compatible with f. Let us write N_{Y}X for the generalized normal bundle of Y in X, i.e., N_{Y}X = Coker(T_{Y/S} → f^{∗}T_{X /S}). Choose a local coordinate patch (f^{−1}(V), t_{1}, …, t_{r}, y_{1} = y_{1} ∘ f, …, y_{m} = y_{m} ∘ f) as above. Write the operator D(∂/∂t_{1}) in the form

D(\frac{∂}{∂t_{1}} ) = f + \frac{∂}{∂t_{1}} + ∑_{i=1}^{c} f_{i} \frac{∂}{∂x_{i}} + ∑_{i=1}^{m} g_{i} \frac{∂}{∂y_{i}} + ∑_{i≤j=1}^{m} f_{ij} \frac{∂^{2}}{∂y_{i}∂y_{j}} + ∑_{i=1}^{c} x_{i} Ξ_{i},

for certain local functions f, f_{i}, g_{i}, f_{ij} and second order operators Ξ_{i}. It can be seen (and we leave this to the reader) that the class of ∑ f_{i} \frac{∂}{∂x_{i}} in N_{Y}X (f^{−1}(V)) is independent of the choice of the local trivialization and coordinates. The upshot of this is that if D is compatible with Y → X, then there is a first order symbol

σ_{f,D} : q^{∗}T_{S} ⟶ N_{Y}X .

We say that D is strictly compatible with f : Y → X if D is compatible with f and the symbol σ_{f,D} is zero. We remark that this notion is well defined for a projective heat operator as well.

Assume that D is strictly compatible with Y → X. It follows from the local description of our compatibility of restrictions that D induces a heat operator D_{Y} on the invertible O_{Y}-module f^{∗}L over S. It is characterized by the property D(θ)(s)|_{Y} = D_{Y}(θ)(s|_{Y}) for any local section θ of T_{S} and any local section s of L. It is clear that the symbol of D_{Y} is equal to ρ_{f,D}, hence that D_{Y} is compatible with D. We say that a heat operator D′ on f^{∗}L is strictly compatible with D if D is strictly compatible with f : Y → X and D′ = D_{Y}. In this case the restriction map

p_{∗}L ⟶ q_{∗}f^{∗}L

is horizontal for the connections induced by D and D′ = D_{Y}. Note D_{Y} is flat if D is flat. Similar remarks hold for projective heat operators and projective flatness.

2.4 Moduli of bundles

2.4.1 The symbol ρ_{Hitchin} in terms of moduli of bundles

In this subsection we explain how to get a symbol map ρ as in 2.3.7 in the case of the relative moduli space of rank 2 bundles of a family of curves.

Let C be a smooth projective curve over Spec(C), and let E be a stable invertible O_{C}- module of rank 2 with det(E) ≅ O_{C}. This gives a point [E] ∈ M_{C}(C). There are canonical identifications

T_{[E]}M_{C} = H^{1}(C, End_{0}(E)) and T_{[E]}^{∗}M_{C} = H^{0}(C, End_{0}(E) ⊗ Ω_{C}^{1})

where End_{0}(E) denotes the sheaf of endomorphisms of E with trace zero. A cotangent vector ϕ ∈ T_{[E]}^{∗}M_{C} thus corresponds to a homomorphism of O_{C}-modules ϕ : E → E ⊗ Ω_{C}^{1}. (Note that this is a Higgs field on the bundle E, see [Si].) Composing and taking the trace gives a symmetric bilinear pairing

\begin{matrix}T_{[E]}^{∗}M_{C} × T_{[E]}^{∗}M_{C} & ⟶ & H^{0}(C, (Ω_{C}^{1})^{⊗2}) \\ (ϕ, ψ) & ⟼ & T race(ϕ ∘ ψ). \end{matrix}

We dualize this and use Serre duality to obtain a map

ρ_{C,E} : H^{1}(C, T_{C}) ⟶ S^{2}T_{[E]}M_{C}.

Let us make the observation that the construction above can be performed in families. Let π : C → S a smooth projective family of curves over the scheme S smooth over Spec(C). Let p : M → S be the associated family of moduli spaces of rank 2 semi-stable bundles with trivial determinant up to S-equivalence. We remark that p is a flat projective morphism, not smooth in general. However, the open part of stable bundles M^{s} ⊂ M is smooth over S. We denote this smooth morphism by p^{s} : M^{s} → S.

Now let T → S be a morphism of finite type. We denote by an index _{T} base change to T. Let E be a locally free sheaf of rank 2 on C_{T}. For a point 0 ∈ T (C) we put C = C_{0} and E = E_{0} = E|_{C}. Suppose that for any 0 we have (a) det(E) ≅ O_{C}, and (b) the bundle E is stable. In this case we get an S-morphism t : T → M^{s}. Analogously to the above we have a canonical isomorphism

t^{∗}(Ω_{M^{s}/S}^{1}) ≅ (π_{T})_{∗}(End_{0}(E) ⊗_{O_{C_{T}}} Ω_{C_{T} /T}^{1} ).

Using the same pairing and dualities as above we get an O_{T} -linear map

R^{1}π_{∗}T_{C_{T} /T} ⟶ S^{2}(t^{∗}(T_{M^{s}/S}^{1})).

If we compose this with the pullback to T of the Kodaira-Spencer map T_{S} → R^{1}π_{∗}T_{C/S} of the family C → S, then we get a map

ρ_{E/C_{T}} : T_{S} ⊗ O_{T} ⟶ S^{2}(t^{∗}(T_{M^{s}/S}^{1})).

If there existed a universal bundle E^{univ} over M^{s}, then we would get a “symbol”

ρ_{Hitchin}^{s} = ρ_{E^{univ}} : (p^{s})^{∗}T_{S} → S^{2}T_{M^{s}/S}.

Although the universal bundle does not exist, the symbol ρ_{Hitchin}^{s} does: it is the unique O_{M^{s}}- linear map such that, whenever E/C_{T} is given, the pullback of ρ_{Hitchin} to T agrees with ρ_{E/C_{T}}^{s}. Uniqueness and existence follows from the existence of “universal” bundles étale locally over M^{s} and the invariance of ρ_{E/C_{T}} under automorphisms of E over C_{T}.

2.4.2 Hitchin’s results in genus at least 3

Let us get back to our curve C over Spec(C). We see from the above that any deformation C_{ϵ} of the curve will give a tensor in Γ(M_{C}^{s}, S^{2}T_{M_{C}}). However, since M_{C} has points which do not correspond to stable bundles (and these points are singular points of M_{C} for g > 2) such a deformation does not give a global tensor in H^{0}(M_{C}, S^{2}T_{M_{C}}). In [Hi, Section 5], one can find a sketch of a proof that the tensor extends if g > 2 (or rank > 2, a case we do not even consider, although our present considerations work in that case as well).

In the situation C → S there is a canonical line bundle L on M; it can be defined by the formula

L = det R(pr_{2})_{∗}E^{univ}, pr_{2} : C ×_{S} M → M.

It is shown in [Hi, Theorem 3.6, Section 5], that if g > 2 the tensor 2/(2k + λ)ρ_{Hitchin} is the symbol of a unique projective flat heat operator D̄_{Hitchin} on L^{⊗k} over S. (We remark that the extra factor 2 comes from our way of normalizing symbols, see Subsection 2.2.4.) In fact the arguments of [Hi] prove the existence of this heat operator over the moduli space of curves of genus g > 2.

We will not use these results.

2.5 The case of genus two curves

2.5.1 Notation in genus 2 case

We fix a scheme S smooth over Spec(C) and a smooth projective morphism π : C → S, whose fibres C are curves of genus 2. We write 0 ∈ S(C) for a typical point and C = C_{0}.

We introduce the following objects associated to the situation. P ic^{1} = P ic_{C/S}^{1} → S denotes the Picard scheme of invertible O_{C} bundles of relative degree 1 on C over S. There is a natural morphism C → P ic^{1} over S, given by P ∈ C(C) maps to [O_{C}(P)] in (P ic^{1})_{0}(C) = P ic^{1}(C). The image of this morphism is a relative divisor Θ^{1} = Θ_{C/S}^{1} on P ic^{1} over S; of course Θ^{1} ≅ C as S-schemes. Let us denote α : P ic^{1} → S the structural morphism. Note that α_{∗}O_{P ic}^{1}(2Θ^{1}) is a locally free O_{S}-module of rank 4 on S. We put P = |2Θ^{1}| equal to the projective space of lines in this locally free sheaf. More precisely, we define

p : P = P (Hom_{O_{S}} (α_{∗}O_{P ic}^{1}(2Θ^{1}) , O_{S})) ⟶ S,

see [Ha, page 162] for notation used. For a point 0 ∈ S(C) we put P = P_{0}.

Let T → S be a morphism of finite type, and let E be a locally free sheaf of O_{C_{T}} -modules of rank 2. For a point 0 ∈ T (C) we put C = C_{0} and E = E_{0} = E|_{C}. Suppose that for any 0 we have

(a) det(E) ≅ O_{C},

(b) the bundle E is semi-stable.

(Compare with Subsection 2.4.1.) We recall the relative divisor D_{E} ⊂ P ic_{T}^{1} associated to E. Set-theoretically it has the following description:

D_{E} := D_{E} ∩ P ic^{1}(C) = {[L] ∈ P ic^{1}(C) : dim H^{1}(C, E ⊗ L) ≥ 1}

To construct D_{E} as a closed subscheme we may work étale locally on T. Hence we may assume that a Poincaré line bundle L on C_{T} ×_{T} P ic_{T}^{1} exists, i.e., which has relative degree 1 for p_{2} : C_{T} ×_{T} P ic_{T}^{1} → P ic_{T}^{1} and induces id : P ic_{T}^{1} → P ic_{T}^{1} . We consider the line bundle

N = (detR(p_{2})_{∗}(p_{1}^{∗}(E) ⊗ L))^{⊗−1}

on P ic_{T}^{1} . This line bundle has a natural section (the theta function) θ(E), which extends the section 1 on the open schematically dense subscheme U ⊂ P ic_{T}^{1} over which the complex R(p_{2})_{∗}(p_{1}^{∗}(E) ⊗ L) is trivial. We define D_{E} to be the zero set of the section θ(E); it is also the

largest closed subscheme of P ic_{T}^{1} over which the coherent O_{P ic}_{T}^{1} -module R^{1}(p_{2})_{∗}(p_{1}^{∗}(E) ⊗ L) has rank ≥ 1 (i.e., defined in terms of a fitting ideal of this sheaf).

One can prove that N ⊗O(−2Θ_{T}^{1} ) is isomorphic to the pullback of an invertible O_{T} -module on T. Hence the divisor D_{E} defines an S-morphism of T into P:

φ_{(T,E)} : T ⟶ P.

These constructions define therefore a transformation of the stack of semi-stable rank 2 bundles with trivial determinant on C over S towards the scheme P. It turns out that this defines an isomorphism of the coarse moduli scheme

M_{C} ⟶ P

towards P, see [NR].

A remark about the natural determinant bundle L on M_{C} (see [Hi, page 360]). The relative Picard group of P over S is Z and is generated by O_{P} (1). Hence it is clear that L ≅ O_{P} (n) ⊗ p^{∗}(N) for some invertible O_{S}-module N on S. The integer n may be determined as follows. We know by [Hi, page 360] that O_{P} (n)^{−λ} ≅ K_{P}. The integer λ = 4 in this case, hence n = 1. We are going to define a projective heat operator on L^{⊗k} over S, but we have seen in Subsection 2.3.5 that this is the same as defining a projective heat operator on O_{P} (k). Hence we will work with the line bundle O_{P} (k) from now on. (Note that as both O_{P} (1) and L are defined on the moduli stack, they must be related by a line bundle coming from the moduli stack M_{2} of curves of genus 2; however P ic(M_{2}) is rather small.)

Let P ic^{0} = P ic_{C/S}^{0} → S denote the Picard scheme of invertible O_{C} bundles of relative degree 0 on C over S. There is a natural S-morphism

f : P ic^{0} ⟶ P

which in terms of the moduli-interpretations of both spaces can be defined as follows:

P ic^{0}(C) ∋ [L] ⟼ [L ⊕ L^{−1}] ∈ P (C).

It is clear that the divisor D_{E} associated to E = L ⊕ L^{−1} is equal to D_{E} = (Θ_{C}^{1} + [L]) ∪ (Θ_{C}^{1} + [L^{−1}]) ⊂ P ic^{1}(C). This implies readily that f^{∗}O_{P} (1) ≅ O_{P ic}^{0}(2Θ^{0}), in fact f^{∗}O_{P} (1) may serve as the definition of O_{P ic}^{0}(2Θ^{0}) on P ic^{0}. This induces an isomorpism P ≅ |O_{P ic}^{0}(2Θ^{0})|^{∗} which identifies f with the natural map. Thus f factors over K := P ic^{0}/⟨±1⟩, the relative Kummer surface, and defines a closed immersion (over S):

K ↪ P.

Any E on C that is semi-stable but not stable is an extension of the form 0 → L → E → L^{−1} → 0, with L of degree zero. This bundle is S-equivalent to the bundle L ⊕ L^{−1}. Thus the open subscheme P^{s} ⊂ P is equal to the complement of the image of f : P ic^{0} → P.

2.5.2 Automorphisms

The group scheme H = P ic^{0}[2] of 2-torsion points of P ic^{0} over S acts on the schemes P ic^{1}, Indeed, if E over C_{T} is a family of locally free sheaves as above and if A is a line bundle on C_{T} which defines a 2-torsion point of P ic^{0}, then E ⊗ A is another family of locally free sheaves satisfying (a) and (b) of 2.5.1. This defines an action H ×_{S} P → P of H on P, given the identification of P as the moduli scheme. It is easy to see that this action is induced from the natural action of H on P ic^{1}. Furthermore the morphism f : P ic^{0} → P is equivariant with respect to this action.

Note that the action just defined does not lift to an action of H on O_{P} (1). Let us write G for theta group of the relatively ample line bundle O_{P ic}^{0}(2Θ^{0}) on P ic^{0} over S; this group scheme gives the Heisenberg group for any point 0 ∈ S(C). We remark that G fits into the exact sequence

1 ⟶ G_{m,S} ⟶ G ⟶ H ⟶ 0.

There is a unique action of G on O_{P} (1) which lifts the action of H on P and agrees with the defining action G on f^{∗}O_{P} (1) over P ic^{0}.

2.5.3 Test families

Let E be a rank 2 free O-module on C_{T} satisfying (a) and (b) of 2.5.1. We will say that the pair (T, E) is a test family if the following conditions are satisfied:

1. T → S is smooth,

2. all E = E_{0} are simple vector bundles on C = C_{0}, and

3. the map T_{T/S} ⟶ R^{1}pr_{2}_{∗}(End_{0}(E)) is an isomorphism.

The last condition needs some clarification. The obstruction for the locally free sheaf E to have a connection on C_{T} is an element (the Atiyah class) in

H^{1}(C ×_{S} T, Ω_{C×_{S}T}^{1} ⊗ End(E)).

We can use the maps Ω_{C×_{S}T}^{1} → pr_{2}^{∗}Ω_{T/S}^{1} and End(E) → End_{0}(E) to project this to a section of Ω_{T/S}^{1} ⊗ R^{1}pr_{2}_{∗}(End_{0}(E)). Whence the map of condition 3. This condition means that the deformation of E = E_{0} for any 0 ∈ T is versal in the “vertical direction”.

The following lemma is proved in the usual manner, using deformation theory (no obstructions !) and Artin approximation.

2.5.4 Lemma. For any 0 ∈ S(C) and any simple bundle E on C = C_{0} there exists a test family (T, E) such that (E, C) occurs as one of its fibres.

2.5.5 Lemma. Let (T, E) be a test family. The morphism φ_{(T,E)} : T → P is étale.

Proof. Let E be a simple rank 2 bundle on a smooth projective genus two curve C over Spec(C) with det(E) = O_{C}. Let η ∈ H^{1}(C, End_{0}(E)). For each invertible sheaf L on C, with deg(L) = −1 and given embedding L ⊂ E consider the map

End_{0}(E) ⟶ Hom(L, E).

We have to show (∗): If for all L ⊂ E as above η maps to 0 in H^{1}(C, Hom(L, E)), then η = 0.

Indeed, the condition η ↦ 0 means that the trivial deformation L[ϵ] of L over C[ϵ] = C × Spec(C[ϵ]) can be embedded into the deformation of E given by η. If this holds for all L ⊂ E, then the divisor D_{E} does not move, and we want this to imply that the infinitesimal deformation of E is trivial.

Let us prove (∗). We will do this in the case that E is not stable, as we already have the result in the stable case, see Subsection 2.4.1. Thus we may assume the bundle E is simple but not stable: E is a nontrivial extension

0 ⟶ A ⟶ E ⟶ A^{−1} ⟶ 0,

with deg_{C}A = 0, A^{⊗2} ≅ O̸_{C}. In this case it is easy to see that there exist three line bundles L_{1} ⊂ E, L_{2} ⊂ E and L_{3} ⊂ E of degree −1 on C such that the arrow towards E in the following exact sequence is surjective:

0 ⟶ K ⟶ L_{1} ⊕ L_{2} ⊕ L_{3} ⟶ E ⟶ 0.

Here K is just the kernel of the surjection. For example we can take L_{i} of the form A^{−1}(−P_{i}) for i = 1, 2 and L_{3} of the form A(−P_{3}). (This is the only thing we will need in the rest of the argument; it should be easy to establish this in the case of a stable bundle E also.) This property then holds for L_{i} ⊂ E sufficiently general also; choose L_{i} sufficiently general. Note that K ≅ L_{1} ⊗ L_{2} ⊗ L_{3} as det(E) = O_{C}.

Suppose we have a deformation E_{ϵ} of E given by η as above, i.e., we have L_{i}[ϵ] → E_{ϵ} lifting L → E. This determines an exact sequence

0 ⟶ K_{ϵ} ⟶ L_{1}[ϵ] ⊕ L_{2}[ϵ] ⊕ L_{3}[ϵ] ⟶ E_{ϵ} ⟶ 0.

But detE_{ϵ} ≅ O_{C}[ϵ], hence K_{ϵ} ≅ L_{1}[ϵ] ⊗ L_{2}[ϵ] ⊗ L_{3}[ϵ] ≅ (L_{1} ⊗ L_{2} ⊗ L_{3})[ϵ] is trivial as well. Now note that by our general position we have

(1)  dim H^{0}(C, Hom(K, L_{1})) = dim H^{0}(C, L_{2} ⊗ L_{3}) = 1.

Similar for L_{2} and L_{3}. Therefore, up to a unit in C[ϵ], there is only one map K[ϵ] → L_{i}[ϵ], lifting K → L_{i}. Thus the sequence (1) above is uniquely determined up to isomorphism, hence E_{ϵ} ≅ E[ϵ] is constant, i.e., η = 0. QED

2.6 Results

This lemma completes the preparations. In the remainder of this section we prove the desired results on the existence and the characterization of the connection.

2.6.1 Theorem. The Hitchin symbol ρ_{Hitchin}^{s} : p^{∗}T_{S} → S^{2}T_{P^{s}/S} defined in Subsection 2.4.1 extends to a symbol

ρ_{Hitchin} : p^{∗}T_{S} ⟶ S^{2}T_{P/S}.

Proof. Let (T, E) be a test family. We have the morphism φ = φ_{(T,E)} : T → P which is étale and which induces therefore an isomorphism T_{T/S} → φ^{∗}T_{P/S}. On the other hand we have the isomorphism T_{T/S} → R^{1}pr_{2}_{∗}(End_{0}(E)) of the test family (T, E). We also have the pairing as in Subsection 2.4.1

pr_{2}_{∗}(End_{0}(E) ⊗ Ω_{C_{T} /T}^{1} ) ⊗ pr_{2}_{∗}(End_{0}(E) ⊗ Ω_{C_{T} /T}^{1} ) ⟶ pr_{2}_{∗}((Ω_{C_{T} /T}^{1} )^{⊗2}).

Note that pr_{2}_{∗}(End_{0}(E) ⊗ Ω_{C_{T} /T}^{1} ) is a locally free sheaf of O_{T} -modules as E has simple fibres. These maps and duality give us together with the Kodaira-Spencer map a map

T_{S} ⊗ O_{T} ⟶ S^{2}φ^{∗}T_{P/S}.

We remark that on the (nonempty) stable locus T^{s} = φ^{−1}(P^{s}) this map is equal to the pullback of ρ_{Hitchin}^{s}.

Thus, by Lemma 2.5.5, we see that we can extend ρ_{Hitchin}^{s} to any point x ∈ P (C) which is the image of some simple bundle. Note that a semi-stable bundle E, which is a nontrivial extension 0 → L → E → L^{−1} → 0 with deg(L) = 0 is simple if and only if L^{⊗2} ≅ O̸_{C}. It follows from Lemma 2.5.4 and the above that we can extend ρ_{Hitchin}^{s} to the complement of the codimension 3 locus f(P ic^{0}[2]), and hence by Hartog’s theorem it extends. QED

2.6.2 Proposition. Let f : P ic^{0} → P be as in 2.5.1, then f is a submersion on the locus Y := P ic^{0} \ P ic^{0}[2]. There is a commutative diagram

f^{∗}(ρ_{Hitchin}) : f^{∗}p^{∗}T_{S} ⟶ f^{∗}S^{2}T_{P/S}
||  ↑
4ρ_{Ab}|_{Y}  : T_{S} ⊗ O_{Y} ⟶ S^{2}T_{Y/S}

where ρ_{Ab} is the symbol of the abelian heat operator on O_{P ic}^{0}(2Θ^{0}) = f^{∗}O_{P} (1).

Proof. Consider once again a simple bundle E, which is given as an extension 0 → L → E → L^{−1} → 0 with deg(L) = 0. Clearly, the kernel of the map

H^{1}(C, End_{0}(E)) ⟶ H^{1}(C, Hom(L, L^{−1}))

represents deformations preserving the filtration on E, i.e., tangent vectors along the Kummer surface. Dually, this corresponds to the image of the map

H^{0}(C, Hom(L^{−1}, L) ⊗ Ω_{C}^{1}) ⟶ H^{0}(C, End_{0}(E) ⊗ Ω_{C}^{1}).

We use the Trace-form to identify End_{0}(E) with its dual. Note that there is a canonical exact sequence

0 ⟶ O_{C} ⟶ End_{0}(E)/L^{⊗2} ⟶ L^{⊗−2} ⟶ 0

induced by the filtration L ⊂ E on E. Note that dim H^{0}(C, (End_{0}(E)/L^{⊗2}) ⊗ Ω_{C}^{1}) = 2, since if it were ≥ 3, then dim H^{0}(C, End_{0}(E) ⊗ Ω^{1}) ≥ 4 contrary to the assumption that E is simple. Thus we get the exact sequence

H^{0}(C, Ω^{1}) ≅ H^{0}(C, (End_{0}(E)/L^{⊗2})⊗Ω_{C}^{1}) ≪⟵ H^{0}(C, End_{0}(E)⊗Ω_{C}^{1}) ↩ H^{0}(C, L^{⊗2}⊗Ω_{C}^{1}) ≅ C.

This is (canonically) dual to the sequence of cotangent spaces

0 ⟶ T_{[L]}P ic^{0} ⟶ T_{f([L])}(P) ⟶ (N_{Y}P)_{[L]} ⟶ 0.

We want to see that our symbols in the point [L] lie in the space S^{2}T_{[L]}P ic^{0}.

We have to study the map

\begin{matrix}H^{0}(C, End_{0}(E) ⊗ Ω_{C}^{1}) × H^{0}(C, End_{0}(E) ⊗ Ω_{C}^{1}) & ⟶ & H^{0}(C, (Ω_{C}^{1})^{⊗2}) \\ X ⊗ η × Y ⊗ ω & ⟼ & T race(XY) ⊗ ηω. \end{matrix}

But the elements of H^{0}(C, L^{⊗2} ⊗ Ω_{C}^{1}), resp. those of H^{0}(C, Ω_{C}^{1}) locally look like

( \begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix} ) ⊗ η, resp. ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ⊗ ω.

Therefore the pairing just reduces to twice the multiplication pairing H^{0}(C, Ω_{C}^{1})×H^{0}(C, Ω_{C}^{1}) → H^{0}(C, (Ω_{C}^{1})^{⊗2}).

This multiplication pairing, however, corresponds excactly (via Kodaira-Spencer) to the symbol of the heat operator on the theta-divisor of the Jacobian P ic^{0} of C over S. Since we are looking at 2Θ^{0}, we get the desired factor 4, compare with Subsections 2.3.8 and 2.3.7. QED

2.6.3 Corollary. The symbol ρ_{Hitchin} is invariant under the action of the group scheme H = P ic^{0}[2] on P and S^{2}T_{P/S}.

Proof. This follows from the observation that it is true for ρ_{Ab|Y} and thus for ρ_{Hitchin}|_{K}, the Hitchin symbol restricted to the Kummer surface. Further, one uses the remark that there are no nonvanishing elements of H^{0}(P, S^{2}T_{P/S}) which vanish on K, that is H^{0}(P, S^{2}T_{P/S}(−4)) = 0. QED

2.6.4 Proposition. There exists a unique projective heat operator D̄_{λ,k} (with λ ∈ C^{∗} be a nonzero complex number) on O_{P} (k) over S with the following properties:

1. the symbol of D̄_{λ,k} is equal to \frac{1}{4λ}ρ_{Hitchin}, and

2. the operator D̄_{λ,k} commutes (projectively) with the action of G on O_{P} (k).

Proof. This follows readily from the discussion in Subsection 2.3.7. Indeed, both obstructions mentioned there vanish in view of the vanishing of R^{1}p_{∗}T_{P/S} and R^{1}p_{∗}O_{P}. We have a good way of choosing the elements D(θ): namely, we choose them G-invariantly. This is possible: first choose arbitrary lifts, and then average over liftings of a full set of sections of H (this can be done étale locally over S). This determines the D(θ) uniquely up to an element of p^{−1}(O_{S}), as (p_{∗}T_{P/S})^{H} = (0). Hence we get the desired projective heat operator. QED

It follows from Proposition 2.6.2 that the heat operators D̄_{λ,k} are compatible with the morphism f : Y → P. Also, for k = 1 and λ = 1, we see that D̄_{1,1} is compatible with the abelian heat operator on O_{P ic}^{0}(2Θ^{0}) restricted to Y (this is one of the reasons for the factor \frac{1}{4}, but see below also).

2.6.5 Hitchin’s connection for genus 2 and rank 2

The title of this subsection is somewhat misleading. As mentioned before, in the paper [Hi] there is no definition of a heat operator in the case of genus 2 and rank 2. However, we propose the following definition.

2.6.6 Definition.

The (projective) Hitchin connection on p_{∗}O_{P} (k) is the projective connection associated to the projective heat operator D_{k} := D̄_{(k+2),k}.

This definition makes sense for the following reason: The symbol of the operator D̄_{(k+2),k} is equal to 2/(2k + 4)ρ_{Hitchin} which is equal to the symbol that occurs in [Hi, Theorem 3.6] (the factor 2 comes from our way of normalizing symbols, see Subsection 2.2.4). Thus the only assumption we needed in order to get D_{k} was the assumption that it is compatible with the action of G, see Subsection 2.6.4.

2.6.7 How to compute the heat operators?

To determine D_{k} we introduce heat operators D_{λ,k} under the following assumptions:

1. The family of Kummer surfaces K ⊂ P is given by one equation F ∈ Γ(P, O_{P} (4)).

2. We have chosen an integrable connection ∇_{0} on F := p_{∗}O_{P} (1) over S which is equivariant for the action of G.

We remark that both conditions can be satisfied on the members of an open covering of S. Also the connection ∇_{0} in 2 is determined uniquely up to the addition of a linear operator of the form η · id for some closed 1-form η on S. We remark that ∇_{0} induces a connection ∇_{0} on all the locally free sheaves S^{k}F = p_{∗}O_{P} (k) over S, but these should not be confused with the connections induced by the D_{λ,k}! In addition ∇_{0} determines an integrable connection on P over S and a lift of this to connections on the sheaves O_{P} (k) over P. In other words we have rigidified (P, O_{P} (k)) over S.

There is a natural surjection

S^{2}F ⊗_{O_{S}} S^{2}F^{∗} ⟶ p_{∗}S^{2}T_{P/S}.

which has a canonical splitting, given by decomposing S^{2}F ⊗_{O_{S}} S^{2}F^{∗} into irreducible GL(F)- modules, i.e., the unique GL(F)-equivariant splitting. Thus we identify a section X of S^{2}T_{P/S} with a section X of S^{2}F ⊗_{O_{S}} S^{2}F^{∗}. Note that we can regard sections of S^{2}F ⊗_{O_{S}} S^{2}F^{∗} as sections of Diff_{P/S}^{(2)}(O_{P} (k)) in a natural manner, by considering them as second order differential operators on the affine 4-space \underline{Spec}(S^{∗}F) invariant under the scalar action of G_{m,S}. We write E for the Euler-vector field, i.e., the section of F ⊗ F^{∗} that corresponds to the identity map of F.

Assume we have a G-invariant section

X ∈ S^{2}F ⊗_{O_{S}} S^{2}F^{∗} ⊗_{O_{S}} Ω_{S}^{1}

with the following property

(Eq.)  X · F = ∇_{0}(F)E in S^{5}(F) ⊗_{O_{S}} F^{∗} ⊗_{O_{S}} Ω_{S}^{1},

where X · F denotes contracting once (note that F can be viewed as a section of S^{4}F).

If we have such an X we define a heat operator

D_{λ,k} : p^{∗}T_{S} ⟶ Diff_{P}^{(2)}(O_{P} (k))

by the formula (θ a local section of T_{S})

D_{λ,k}(θ) = θ − \frac{1}{λ}X_{θ,k}.

Here X_{θ,k} means the following: first contract X with θ to get a section X_{θ} of S^{2}F ⊗_{O_{S}} S^{2}F^{∗} and then consider this as a second order differential operator X_{θ,k} of O_{P} (k) on X over S by the remarks above. Note that the term θ acts on O_{P} (k) using ∇_{0}.

2.6.8 Lemma. Let D_{λ,k} be the heat operator defined above.

1. The heat operator D_{λ,k} commutes with the action of G on O_{P} (k).

2. The heat operators D_{λ,k} are compatible with f : Y → P. If λ = k then D_{k,k} is strictly compatible with f : Y → P.

Proof. The second statement holds since X is G invariant. We now verify the last statement. Note that D_{k,k} is strictly compatible with K → P if the operators D_{k,k}(θ) preserve the subsheaf I_{K} · O_{P} (k) of O_{P} (k). This can be seen by the description of strict compatiblility in terms of local coordinates given in Subsection 2.3.10. We may check this condition on the affine 4-space \underline{Spec}(S^{∗}F). Thus let G ∈ S^{k−4}(F) be a section, let θ be a local vector field on S and consider

∇_{0,θ}(GF) − \frac{1}{k}X_{θ}(GF)

We have to show that this is divisible by F.

We may assume that G is horizontal for ∇_{0}, as we can find a horizontal basis for S^{k−4}(F) locally. Thus we get

G∇_{0,θ}(F) − \frac{1}{k}X_{θ}(GF) = G∇_{0,θ}(F) − \frac{1}{k}X_{θ}(G)F − \frac{1}{k}(X · F)_{θ}(G) − \frac{1}{k}GX_{θ}(F).

Here we have used a general formula for the application of a second order operator like X_{θ} on a product like F G. Note also that X(F) is simply the contraction of X · F, hence by (Eq.) we get

G∇_{0,θ}(F) − \frac{1}{k}X_{θ}(G)F − \frac{1}{k}∇_{0,θ}(F)E(G) − \frac{1}{k}GE(∇_{0,θ}(F)).

Next, we use that acting by E on a homogeneous polynomial gives degree times the polynomial:

G∇_{0,θ}(F) − \frac{1}{k}X_{θ}(G)F − \frac{1}{k}∇_{0,θ}(F)(k − 4)G − \frac{1}{k}G4∇_{0,θ}(F) = −\frac{1}{k}X_{θ}(G)F

which is divisible by F. Thus D_{k,k} is indeed stricly compatible with f : Y → P. QED 2.6.9 Theorem. Let X be a solution to the equation (Eq.) above and let D_{λ,k} be the heat operators defined in this subsection. Then

1. The projective heat operator defined by D_{λ,k} is the operator D̄_{λ,k} defined in Subsection 2.6.4. Thus D_{k+2,k} defines the Hitchin connection on O_{P} (k).

2. The heat operators D_{λ,k} are projectively flat for any k and λ.

3. For λ = k, the operator D_{k,k} is strictly compatible with the abelian projective heat operator on O_{P ic}^{0}(O(2kΘ^{0})) over an open part of P ic^{0}.

Proof. Consider first the case of D_{k,k} as defined in 2.6.7. We have seen above that it is strictly compatible with f : Y → P. Therefore, by Subsection 2.3.10 it defines a heat operator D_{k} on f^{∗}O_{P} (k) over Y. In view of Hartog’s theorem this extends uniquely to a heat operator D_{k} of O_{P ic}^{0}(O(2kΘ^{0})) on P ic^{0} over S. We have seen in Subsection 2.3.8 that there is a unique such heat operator, whose symbol is 1/k times the symbol ρ_{Ab} of D_{1}. Combining this with the assertions in Subsection 2.6.2 we see that (1/4k)ρ_{Hitchin} − ρ_{D_{k,k}} vanishes along K and hence is zero. This proves that D_{k,k} agrees with the projective heat operator D̄_{k,k} defined in Subsection 2.6.4.

Now it follows from the transformation behaviour of the symbol of D̄_{λ,k} and D_{λ,k} that these agree for arbitrary λ. Thus we get the agreement stated in the theorem.

To see that these heat operators are projectively flat, we argue as follows. Let θ, θ′ be two local commuting vector fields on S. We have to see that the section [D_{λ,k}(θ), D_{λ,k}(θ′)] of Diff_{P/S}^{(3)}(O_{P} (k)) lies in the subsheaf p^{−1}(O_{S}). Again, for λ = k this section is zero when restricted to K, and again this implies that the section is zero in that case. Now, let us compute

[D_{λ,k}(θ), D_{λ,k}(θ′)] = [θ − \frac{1}{λ}X_{θ,k}, θ′ − \frac{1}{λ}X_{θ′,k}] = \frac{1}{λ}([θ, X_{θ′,k}] − [θ′, X_{θ,k}]) + \frac{1}{λ^{2}} [X_{θ,k}, X_{θ′,k}].

The index _{k} is now superfuous, as we can see this as an expression in S^{2}F ⊗_{O_{S}} S^{2}F^{∗} ⊕S^{3}F ⊗_{O_{S}} S^{3}F^{∗}. We know that this expression is zero for any k with λ = k considered as a third order operator on O_{P} (k). This implies that both terms are zero, hence the expression is zero for any λ and any k. (We remark for the doubtful reader that we will verify the flatness also by a direct computation using the explicit description of the operator.) QED

3 The flat connection

3.1

We introduce certain families of flat connections on (trivial) bundles over the configuration space P (cf. 1.4). These are defined by representations of the Lie algebra so(2g + 2). Then we derive a convenient form for the equation of the family of Kummer surfaces K ↪ P (in fact for any g we point out a specific section P_{z} of a trivial bundle over P). In Theorem 3.4.3 we show that this equation is flat for one of our connections. This will be important in identifying Hitchin’s connection in the next section.

3.2 Orthogonal groups

3.2.1 The Lie algebra

Let Q = x_{1}^{2} + … + x_{2g+2}^{2}, then the (complex) Lie algebra so(Q) = so(2g + 2) is:

so(2g + 2) = {A ∈ End(C^{2g+2}) : ^{t}A + A = 0 }, let F_{ij} := 2(E_{ij} − E_{ji}) (∈ so(2g + 2)),

where E_{ij} is the matrix whose only non-zero entry is a 1 in the (i, j)-th position (the commutator of such matrices is [E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{li}E_{kj}). The alternating matrices F_{ij} (with 1 ≤ i < j ≤ 2g + 2) are a basis of so(2g + 2). Note F_{ij} = −F_{ji}. These matrices satisfy the relations:

[F_{ij}, F_{kl}] = \{ \begin{matrix}0 & if & i, j, k, l are distinct, \\ 2F_{il} & if & j = k. \end{matrix}

3.2.2 The universal envelopping algebra

Recall that for a Lie algebra g the tensor algebra T (g) and the universal enveloping algebra U(g) are defined by:

T (g) := C ⊕ g ⊕ g ⊗ g …, U(g) := T (g)/I

where I is the ideal generated by all elements of the form x ⊗ y − y ⊗ x − [x, y] with x, y ∈ g. Lie algebra representations ρ : g → End(V) (so ρ([x, y]) = ρ(x)ρ(y) − ρ(y)ρ(x) correspond to representations ρ̃ : U(g) → End(V) of associative algebras with ρ(1) = id_{V}. Given ρ one defines ρ̃(x_{i_{1}} ⊗ x_{i_{2}} … ⊗ x_{i_{k}}) := ρ(x_{i_{1}})ρ(x_{i_{2}}) … ρ(x_{i_{k}}) which is well-defined because we work modulo I.

3.2.3 Definitions

We define elements in U(so(2g + 2)) by:

Ω_{ij} := F_{ij} ⊗ F_{ij} mod I  (∈ U(so(2g + 2))),  note Ω_{ij} = Ω_{ji}.

In particular, for any Lie algebra representation ρ : so(2g+2) → W we now have endomorphisms ρ̃(Ω_{ij}) = ρ(F_{ij})^{2} : W → W which satisfy the same commutation relations as the Ω_{ij}.

For λ ∈ C^{∗} we define an U(so(2g + 2))-valued one form ω_{λ} on P:

ω_{λ} := λ^{−1} (∑_{i≠j} \frac{Ω_{ij}dz_{i}}{z_{i} − z_{j}} ); let ρ̃(ω_{λ}) := λ^{−1} (∑_{i≠j} \frac{ρ̃(Ω_{ij})dz_{i}}{z_{i} − z_{j}} )

where ρ̃: U(so(2g + 2)) → End(W) is a representation. Note ρ̃(ω_{λ}) ∈ End(W) ⊗ Ω_{P}^{1}. The one form ω_{λ} defines a connection on the trivial bundle W ⊗_{C} O_{P} by:

∇_{λ}^{W} : W ⊗ O_{P} ⟶ W ⊗ Ω_{P}^{1},  fw ⟼ w ⊗ df − fρ̃(ω_{λ})(w)  (f ∈ O_{P}, w ∈ W).

The covariant derivative of f ⊗ w with respect to the vector field ∂/∂z_{i} on P is the composition of ∇_{λ}^{W} with contraction:

(∇_{λ}^{W} (f ⊗ w))_{∂/∂z_{i}} = w ⊗ \frac{∂f}{∂z_{i}} − ∑_{j, j≠i} ρ̃(Ω_{ij})(w) ⊗ \frac{1}{z_{i} − z_{j}} (∈ W ⊗ O_{P}).

3.2.4 Proposition. Let ρ : so(2g + 2) ⟶ End(W) be a Lie algebra representation, then ∇_{λ}^{W} is a flat connection on W ⊗ O_{P} for any λ ∈ C^{∗}.

Proof. Obviously ∇_{λ}^{W} is a connection. It is well known (and easy to verify) that the connection defined by ω is flat (so dω + ω ∧ ω = 0) if the following infinitesimal pure braid relations in U(so(2g + 2)) are satisfied:

[Ω_{ij}, Ω_{kl}] = 0, [Ω_{ik}, Ω_{ij} + Ω_{jk}] = 0,

where i, j, k, l are distinct indices (cf. [Ka], Section XIX.2).

To check these relations we use that X ⊗ Y = Y ⊗ X + [X, Y] in U(so). The first relation is then obviously satisfied. We spell out the second. Consider first:

F_{ik} ⊗ F_{ij} ⊗ F_{ij} = (F_{ij} ⊗ F_{ik} + [F_{ik}, F_{ij}]) ⊗ F_{ij}
= F_{ij} ⊗ F_{ik} ⊗ F_{ij} − F_{kj} ⊗ F_{ij}
= F_{ij} ⊗ (F_{ij} ⊗ F_{ik} − F_{kj}) + F_{jk} ⊗ F_{ij}
= F_{ij} ⊗ F_{ij} ⊗ F_{ik} + (F_{ij} ⊗ F_{jk} + F_{jk} ⊗ F_{ij}),

so we have [F_{ik}, F_{ij} ⊗ F_{ij}] = D_{1} with D_{1} symmetric in the indices i and k. Similarly we get:

F_{ik} ⊗ (F_{ik} ⊗ F_{ij} ⊗ F_{ij}) = F_{ik} ⊗ (F_{ij} ⊗ F_{ij} ⊗ F_{ik} + D_{1})
= (F_{ik} ⊗ F_{ij} ⊗ F_{ij}) ⊗ F_{ik} + F_{ik} ⊗ D_{1}
= F_{ij} ⊗ F_{ij} ⊗ F_{ik} ⊗ F_{ik} + (D_{1} ⊗ F_{ik} + F_{ik} ⊗ D_{1}).

Thus [Ω_{ik}, Ω_{ij}] = D_{1} ⊗ F_{ik} + F_{ik} ⊗ D_{1} which is antisymmeric in i and k. Therefore

[Ω_{ik}, Ω_{ij}] = −[Ω_{ki}, Ω_{kj}] = −[Ω_{ik}, Ω_{jk}]

which proves the second infinitesimal braid relation. QED

3.2.5 Differential operators

Given a Lie algebra representation ρ : g → End(S_{1}) where S_{k} ⊂ S := C[…, X_{i}, …] is the subspace of homogeneous polynomials of degree k, there is a convenient way to determine the representation ρ^{(k)} : g → End(S_{k}) induced by ρ. If ρ(A) = (a_{ij}) w.r.t. the basis X_{i} of S_{1}, then we define

L_{A} := ∑_{i,j} a_{ij}X_{i}∂_{j}  (∈ End(S)),  with ∂_{j}(P) := \frac{∂P}{∂X_{j}}

for P ∈ S. Then obviously ρ(A)(P) = L_{A}(P) if P is linear and the Leibnitz rule shows that ρ^{(k)}(A) = L_{A} : S_{k} → S_{k} for all k.

The composition (in End(S)) of two operators is given by:

(X_{i}∂_{j}) ∘ (X_{k}∂_{l}) = X_{i}X_{k}∂_{j}∂_{l} + δ_{jk}X_{i}∂_{l}, thus L_{A}^{2} = (1/2)σ(L_{A}^{2}) + L_{A^{2}},

with symbol σ(L_{A}^{2}) = 2 ∑_{i,j,k,l} a_{ij}a_{kl}X_{i}X_{k}∂_{j}∂_{l} (see our convention 2.2.4).

Assume that we have A_{ij} ∈ g such that Ω_{ij} = A_{ij}⊗A_{ij} satisfy the infinitesimal braid relations (cf. proof of the previous proposition) and that in the representation ρ we have ρ(A_{ij})^{2} = μI with μ ∈ C. Then we conclude that the operators

σ(L_{A_{ij}}^{2}) : S ⟶ S

also satisfy the infinitesimal braid relations.

3.3 The Kummer equation.

3.3.1 Introduction

We recall how one can obtain the equation of the Kummer variety for a genus two curve, and more generally, how a hyperelliptic curve C_{z} determines in a natural way a quartic polynomial P_{z} ∈ S^{4}V (ω_{g+1}) where V (ω_{g+1}) is a half spin representation of so(2g + 2).

The polynomial P_{z} lies in the subrepresentation of V (4ω_{g+1}) which also occurs in S^{2}(∧^{g+1}C^{2g+2}) where C^{2g+2} is the standard representation of so(2g + 2). We will exploit this fact to verify that P_{z} is a flat section for one of the connections introduced in Subsection 3.2.

3.3.2 The orthogonal Grassmanian

We define two quadratic forms on C^{2g+2} by:

Q : x_{1}^{2} + … + x_{2g+2}^{2}, Q_{z} : z_{1}x_{1}^{2} + … + z_{2g+2}x_{2g+2}^{2},

and we use the same symbols to denote the corresponding quadrics in P^{2g+1}. The quadric Q has two rulings (families of linear P^{g}’s lying on it). Each of these is parametrized by the orthogonal Grassmanian (spinor variety) denoted by Gr_{SO}. The variety Gr_{SO} is smooth, projective of dimension \frac{1}{2}g(g + 1).

Let V (ω_{g+1}) be a half spin representation of so(2g + 2). There is an embedding

ϕ : Gr_{SO} ⟶ P V (ω_{g+1})^{∗}, let O_{Gr_{SO}}(1) := ϕ^{∗}O_{P V} (1).

in fact, ϕ(Gr_{SO}) is the orbit of the highest weight vector. The map ϕ is equivariant for the action of the spin group S̃O (which acts through a half-spin representation on V (ω_{g+1})). It induces isomorphisms:

H^{0}(Gr_{SO}, O(n)) = V (nω_{g+1}).

In case g = 2, one has isomorphisms:

ϕ : Gr_{SO} ⟶^{≅} P^{3}, S_{k} = V (kω_{g+1}) and S̃O_{6} ≅ SL_{4}(C)

and the halfspin representation is identified with the standard representation SL_{4}(C) on C^{4} (or its dual).

3.3.3 The Plücker map

Let Gr_{SL} be the Grassmanian of g-dimensional subspaces of P^{2g+1}, we denote the Plücker map by:

p : Gr_{SL} ⟶ P ∧^{g+1} C^{2g+2}, ⟨v_{1}, …, v_{g+1}⟩ ⟼ v_{1} ∧ … ∧ v_{g+1} = ∑ p_{i_{1}...i_{g+1}}e_{i_{1}} ∧ … ∧ e_{i_{g+1}}.

The map p is equivariant for the SL(2g + 2)-action on both sides and induces isomorphisms of sl(2g + 2)-representations

H^{0}(Gr_{SL}, O_{Gr_{SL}}(n)) = V (nλ_{g+1}), with V (λ_{g+1}) ≅ ∧^{g+1}C^{2g+2}.

As a representation of so(2g + 2) the space ∧^{g+1}C^{2g+2} is reducible, one has (cf. [vG] (3.7), [FH], §19.2, Th. 19.2 and Remarks (ii)), with V (ω_{g}) and V (ω_{g+1}) the half spin representations:

∧^{g+1}C^{2g+2} = V (2ω_{g}) ⊕ V (2ω_{g+1}).

In particular, the irreducible component V (4ω_{g+1}) of S^{4}V (ω_{g+1}) is also a component of S^{2}(∧^{g+1}C^{2g+2}) (viewed as so(2g + 2) representation). Elements of S^{2}(∧^{g+1}C^{2g+2}) can be seen as quadratic forms in the Plücker coordinates, when restricted to Gr_{SO} they can be viewed as (restrictions to Gr_{SO} of) quartic polynomials on P V (ω_{g+1})^{∗}.

3.3.4 Notation

The following lemma gives this decomposition of ∧^{g+1}C^{2g+2} explicitly. Let {e_{i}} be the standard basis of C^{2g+2}. For any

S ⊂ B := {1, 2, …, 2g + 2}, |S| = g + 1,

we write S = {i_{1}, i_{2}, …, i_{g+1}} with i_{1} < i_{2} < … < i_{g+1} and define

e_{S} := e_{i_{1}} ∧ … ∧ e_{i_{g+1}} ∈ ∧^{g+1}C^{2g+2}.

These elements give a basis of ∧^{g+1}C^{2g+2}. For such an S we let S′ := B − S be the complement of S in B. Writing S′ = {j_{1}, j_{2}, …, j_{g+1}} with j_{1} < j_{2} < … < j_{g+1}, we define an element σ_{S} in the symmetric group S_{2g+2} by:

σ_{S}(k) := i_{k}, σ_{S}(g + 1 + k) := j_{k}.

3.3.5 Lemma. The two non-trivial so(2g + 2)-invariant subspaces in ∧^{g+1}C^{2g+2} are:

⟨…, e_{S} + sgn(σ_{S})i^{g+1}e′_{S}, …⟩_{S∋1} and ⟨…, e_{S} − sgn(σ_{S})i^{g+1}e′_{S}, …⟩_{S∋1},

where S runs over the subsets of B with g + 1 elements with 1 ∈ S.

Proof. (Cf. [FH], §19.2 Remarks(iii).) The quadratic form Q defines an so-equivariant isomorphism

B : ∧^{g+1}C^{2g+2} ⟶ (∧^{g+1}C^{2g+2})^{∗} = ∧^{g+1}(C^{2g+2})^{∗}, e_{S} ⟼ ϵ_{S} := ϵ_{i_{1}} ∧ … ∧ ϵ_{i_{g+1}}

with {…, ϵ_{j}, …} the basis dual to the e_{i}. There is also a canonical (in particular so-equivariant) isomorphism:

C : ∧^{g+1}C^{2g+2} ⟶ (∧^{g+1}C^{2g+2})^{∗}, α ⟼ [β ↦ c_{α,β}]

with c_{α,β} ∈ C defined by

α ∧ β = c_{α,β}e_{1} ∧ e_{2} ∧ … e_{2g+2}.

Thus we have an isomorphism A := B^{−1} ∘ C : ∧^{g+1}C^{2g+2} → ∧^{g+1}C^{2g+2} whose eigenspaces are so-invariant. An easy computation shows that the eigenvalues of A are ±i^{g+1} and the eigenvectors are e_{S} ± sgn(σ_{S})i^{g+1}e′_{S}. QED

3.3.6 A quartic polynomial

We consider the g-dimensional subspaces of P^{2g+1} which are tangent to Q_{z}:

B̄_{z} := {P^{g} ⊂ P^{2g+1} : rank(Q_{z} restricted to P^{g}) ≤ g } (⊂ Gr_{SL}).

The subvariety B̄_{z} is defined by a quadratic polynomial P_{z} in the Plücker coordinates, that is by a section in H^{0}(Gr_{SL}, O_{Gr_{SL}}(2)) ([vG], Thm 3):

B̄_{z} = Z(P_{z}), P_{z} ∈ H^{0}(Gr_{SL}, O_{Gr_{SL}}(2)) ⊂ S^{2}(∧^{g+1}C^{2g+2}).

This section is in fact the unique SO_{z}-invariant in H^{0}(Gr_{SL}, O_{Gr_{SL}}(2)), in case g = 2 the quartic surface in P V defined by P_{z} is the Kummer surface of the curve C_{z} ([DR]).

3.3.7 Lemma. The variety B̄_{z} is defined by

P_{z} = ∑_{S} z_{S}e_{S}^{2},

where S runs over the subsets with g + 1 elements of B = {1, …, 2g + 2} and

z_{S} := z_{i_{1}}z_{i_{2}} … z_{i_{g+1}},  e_{S} = e_{i_{1}} ∧ … ∧ e_{i_{g+1}}  if S = {i_{1}, …, i_{g+1}}.

Proof. From [vG], Theorem 3′ and its proof we know that the trivial representation of so_{z} has multiplicity one in H^{0}(Gr_{SL}, O_{Gr_{SL}}(2)) = V (2λ_{g+1}). This sl_{2g+2} representation corresponds to the partition μ: 2g + 2 = 2 + 2 + … + 2 and is thus realized as (C^{2g+2})^{⊗2g+2}c_{μ} where c_{μ} is the Young symmetrizer, one has in fact V (2λ_{g+1}) ⊂ S^{2}(∧^{g+1}C^{2g+2}) (see [FH], §15.5, p. 233-236). We use the standard tableau as in [vG] (and [FH]) to construct c_{μ}.

The trivial so_{z} sub-representation is obtained as follows. The quadratic form Q_{z} defines the so_{z}-invariant tensor ∑_{i=1}^{2g+2} z_{i}e_{i} ⊗ e_{i} ∈ C^{2g+2} ⊗ C^{2g+2}, and we consider the g + 1-fold tensor product of this tensor, where we insert the k-th factor in the positions k and g + 1 + k:

τ_{z} := ∑_{i_{1},...,i_{g+1}=1}^{2g+2} z_{i_{1}}z_{i_{2}} … z_{i_{g+1}}e_{i_{1}} ⊗ e_{i_{2}} ⊗ … ⊗ e_{i_{g+1}} ⊗ e_{i_{1}} ⊗ … e_{i_{g+1}} (∈ (C^{2g+2})^{⊗2g+2}).

Thus τ_{z} is so_{z} invariant and it has the advantage that τ_{z}c_{μ} is easily determined:

τ_{z}c_{μ} = (cst) ∑_{S} z_{S} e_{S} · e_{S},

with (cst) a non-zero integer and where we sum over all subsets S with g + 1 elements of {1, …, 2g+2} (In fact, c_{μ} = a_{μ}b_{μ} and a_{μ} symmetrizes the indices 1, g+2; 2, g+3; …; g+1, 2g+2 of a tensor (but τ_{z} is already symmetric in these indices) and next b_{μ} antisymmetrizes the first g + 1 and last g + 1 indices, giving the result above). QED

3.4 Verifying the contraction criterium

3.4.1

We need to determine the images of the Ω_{ij}’s in the representation S^{2}(∧^{g+1}C^{2g+2}). In the standard representation C^{2g+2} of so(2g + 2), with basis e_{i} and dual basis ϵ_{i}, we have:

F_{ij} = 2(e_{i} ⊗ ϵ_{j} − e_{j} ⊗ ϵ_{i}) ∈ (C^{2g+2}) ⊗ (C^{2g+2})^{∗} = End(C^{2g+2}).

On ∧^{g+1}C^{2g+2}, with basis e_{S} = e_{i_{1}} ∧ … ∧ e_{i_{g+1}} as before, F_{ij} acts as

F_{ij}(e_{i_{1}} ∧ … ∧ e_{i_{g+1}}) = F_{ij}(e_{i_{1}}) ∧ e_{i_{2}} ∧ … ∧ e_{i_{g+1}} + … + e_{i_{1}} ∧ … ∧ e_{i_{g}} ∧ F_{ij}(e_{i_{g+1}}).

Define a subset of B = {1, …, 2g + 2} by

S + ij = (S ∪ {i, j}) − (S ∩ {i, j}).

Then F_{ij}(e_{S}) is, up to sign, 2e_{S+ij} if |S ∩ {i, j}| = 1 and is zero otherwise:

F_{ij} = 2 ∑_{S, |S∩{i,j}|=1} t_{S,ij}e_{S+ij} ⊗ ϵ_{S}

with t_{S,ij} = ±1. Note that if |S ∩ {i, j}| = 1 then F_{ij}^{2}e_{S} = −4e_{S} so t_{S,ij} = −t_{S+ij,ij}.

3.4.2 Lemma. Let W be a complex vector space, let D ∈ W ⊗ W^{∗} be a linear differential operator and let P ∈ S^{k}W. Then the contraction of P with the symbol of D^{2} is given by:

σ(D^{2}) · P = 2D(P)D ∈ S^{k−1}W ⊗ W^{∗}.

Proof. Straightforward using the conventions of 2.2.4 and 3.2.5. QED

3.4.3 Theorem. The composition, still denoted by P_{z}:

P ⟶ S^{2} ∧^{g+1} C^{2g+2} ⟶ S^{4}V (ω_{g+1}), z ⟼ P_{z},

with P_{z} = ∑_{S} z_{S}e_{S}^{2} (as in Subsection 3.3.6), satisfies the differential equations:

(\frac{1}{16} ∑_{j≠i} \frac{σρ̃(Ω_{ij})}{z_{i} − z_{j}} ) · P_{z} + (∂_{z_{i}}P_{z})E = 0 (1 ≤ i ≤ 2g + 2)

with E = ∑ e_{S} ⊗ ϵ_{S} the Euler vector field and ρ̃: U(so(2g + 2)) → End(S^{2}(∧^{g+1}C^{2g+2})). Thus P_{z} is a horizontal section for the connection ∇_{λ}^{W} with W = S^{4}V (ω_{g+1}) and λ = −16.

Proof. For symmetry reasons it suffices to verify the equation for i = 1. By Lemma 3.4.2

σρ̃(Ω_{1j}) · e_{S}^{2} = 2ρ(F_{1j})(e_{S}^{2})ρ(F_{1j}) = 4e_{S}ρ(F_{1j})(e_{S})ρ(F_{1j}).

To simplify notation we write Ω_{ij} for σρ̃(Ω_{ij}) and F_{ij} for ρ(F_{ij}) (as in 3.4.1). Then F_{1j}(e_{S}) = 0 unless |S ∩ {1, j}| = 1. Assume 1 ∈ S and j ∉ S (so j ∈ B − S) and consider the contraction of Ω_{1j} with the term z_{S}e_{S}^{2} + z_{S+ij}e_{S+ij}^{2} from P_{z}:

Ω_{1j} · (z_{S}e_{S}^{2} + z_{S+1j}e_{S+1j}^{2}) = 8(t_{S,1j}z_{S} + t_{S+1j,1j}z_{S+1j})e_{S}e_{S+1j}F_{1j}
= 8t_{S,1j}(z_{1}zS̄ − z_{j}zS̄)e_{S}e_{S+1j}F_{1j}
= (z_{1} − z_{j})Ω_{1j} · zS̄e_{S}^{2},

where S̄ := S − {1}. Therefore we get:

\frac{Ω_{1j}}{z_{1} − z_{j}} · P_{z} = Ω_{1j} · ( ∑_{S∋1, j∉S} zS̄e_{S}^{2}) .

Summing this result over all j and changing the order of summation gives:

\begin{aligned}(∑_{j≠1} \frac{Ω_{1j}}{z_{1}−z_{j}} ) · P_{z} & = & ∑_{S∋1} zS̄ (∑_{j∈B−S} Ω_{1j} · e_{S}^{2}) \\ & = & 8 ∑_{S∋1} zS̄e_{S} (∑_{j∈B−S} t_{S,1j}e_{S+1j}F_{1j}) \end{aligned}

where, as before, B = {1, …, 2g + 2}. On the other hand, since P_{z} = ∑ z_{S}e_{S}^{2}, we have:

∂_{z_{1}}P_{z} = ∑_{S∋1} zS̄e_{S}^{2}.

Thus the theorem follows if we prove, for all S with 1 ∈ S:

2e_{S}E + ∑_{j∈B−S} t_{S,1j}e_{S+1j}F_{1j} = 0.

With the definition of F_{1j} we find:

\begin{aligned}∑_{j∈B−S} t_{S,1j}e_{S+1j}F_{1j} & = & ∑_{j∈B−S} t_{S,1j}e_{S+1j} (∑_{T, |T ∩\{1,j\}|=1} 2t_{T,1j}e_{T +1j} ⊗ ϵ_{T} ) \\ & = & 2 ∑_{T} (∑_{j∈B−S, |T ∩\{1,j\}|=1} t_{S,1j}t_{T,1j}e_{S+1j}e_{T +1j}) ⊗ ϵ_{T} . \end{aligned}

Note that e_{S}E = ∑_{T} e_{S}e_{T} ⊗ ϵ_{T} (with T ∈ B, |T | = g + 1). Comparing coefficients of ϵ_{T}, it remains to prove that (for all S ∋ 1 and all T):

e_{S}e_{T} + ∑_{j∈B−S, |T ∩{1,j}|=1} t_{S,1j}t_{T,1j}e_{S+1j}e_{T +1j} = 0.

We show that the relations for 1 ∈ T follow from those with 1 ∉ T. In fact, we only want this relation in V (4ω_{g+1}), so we restrict ourselves to the subspace V (2ω_{g+1}) ⊂ ∧^{g+1}C^{2g+2}. The kernel of the restriction map is ⟨…, e_{T} − sgn(e_{T})i^{g+1}e_{T′}, …⟩ (or the other space in Lemma 3.3.5; the argument we give leads to the same conlusion in both cases).

Consider a T with 1 ∈ T and j ∉ T. Then 1 ∉ T + 1j and after restriction:

e_{T} = sgn(σ_{T})i^{g+1}e_{T′}, e_{T +1j} = sgn(σ_{(T +1j)′})i^{−(g+1)}e_{(T +1j)′}

where ′ stands for the complement in B. Since (T + 1j)′ = T′ + 1j, substituting these relations and multiplying throughout by i^{g+1} we get:

(−1)^{g+1}sgn(σ_{T})e_{S}e_{T′} + ∑_{j∈B−S, |T ∩{1,j}|=1} t_{S,1j}t_{T,1j}sgn(σ_{T′+1j})e_{S+1j}e_{T′+1j}

Next we observe that

sgn(σ_{T′+1j}) = (−1)^{g+1}t_{T,1j}t_{T′,1j}sgn(σ_{T}),

so it suffices to consider the relations with 1 ∉ T. Let T := {1 = i_{1}, …, i_{g+1}}, T′ := {j_{1}, …, j_{g+1}} with j = j_{k} and i_{l} < j < j_{l+1}. To get σ_{T′+1j}, first apply the permutation (1 g + 2) … (g + 1 2g + 2) to B, the sign of this permutation is (−1)^{g+1}. Then apply σ_{T}. Next apply the cyclic permutation (j_{1} … j_{g+1} 1) (with sign (−1)^{g+1}) and finally apply the inverse of (j_{k} … j_{g+1} i_{2} … i_{l}) (with sign (−1)^{g−k+l}). The resulting permutation is σ_{T′+1j} and has sign (−1)^{g+k+l}sgn(σ_{T}). On the other hand, since 1 ∈ T, j ∉ T we get F_{1j}(e_{T}) = −2(−1)^{l−1}e_{T +1j}, so t_{T,1j} = (−1)^{l} and F_{1j}(e_{T′}) = +2(−1)^{k−1}e_{T′+1j}, so t_{T′,1j} = (−1)^{k−1}. This gives the formula for sgn(σ_{T′+1j}).

From now on we consider only T ’s with 1 ∉ T and S with 1 ∈ S. We show that the desired relation:

e_{S}e_{T} + ∑_{i∈(B−S)∩T} t_{S,1i}t_{T,1i}e_{S+1i}e_{T +1i}.

is a Plücker relation. Thus it holds in H^{0}(Gr_{SL}, O(2)) and therefore also upon restriction to Gr_{SO} ⊂ Gr_{SL}.

Let S = {1 = i_{1}, …, i_{g+1}} and consider the Zariski open subset U ⊂ Gr_{SL} of g + 1dimensional subspaces W ⊂ C^{2g+2} with Plücker coordinate p_{S} ≠ 0. Any such W has a (unique) basis {…, w_{i}, …} with:

W = ⟨w_{1}, …, w_{g+1}⟩,  and (w_{k})_{i_{l}} = δ_{kl},  (S = {1 = i_{1}, …, i_{g+1}}, i_{1} < … < i_{g+1})

where δ_{kl} is Kronecker’s delta. Let M be the (g + 1) × (2g + 2) matrix whose rows are the w_{i}. Let M_{i} be the i-th column of M_{W} (note M_{i_{j}} = f_{j}, the j-th standard basis vector of C^{g+1}). Then the Plücker coordinate p_{k_{1},...,k_{g+1}} is the determinant of the (g + 1) × (g + 1) submatrix of M whose j-th column is M_{k_{j}}, we write:

p_{k_{1},...,k_{g+1}} = det(M_{k_{1}}, …, M_{k_{g+1}}) with k_{1} < … < k_{g+1}.

Let j ∉ S, with i_{l} < j < i_{l+1}. Then p_{S+1j} = (−1)^{l−1}(M_{j})_{1} since

det(f_{i_{2}},…, f_{i_{l}}, M_{j}, f_{i_{l+1}},…, f_{g+1})= (−1)^{l−1}det(M_{j}, f_{i_{2}},…, f_{i_{l}}, f_{i_{l+1}},…, f_{g+1}) =(−1)^{l−1}(M_{j})_{1}.

As earlier, since 1 ∈ S we have t_{S,1j} = −(−1)^{l−1}, hence p_{S+1j} = −t_{S,1j}(M_{j})_{1}.

Let T = {j_{1}, …, j_{g+1}}, let T ∩ S′ = {a_{1}, …, a_{q}} and let j = j_{n} = a_{k}. Then p_{T} = det(M_{j_{1}}, …, M_{j_{g+1}}) is, up to sign, the determinant of a q × q submatrix N of the (g + 1) × q matrix with columns M_{a_{1}}, …, M_{a_{q}}, we write p_{T} = t · det(N) with t = ±1. Then

p_{T +1j} = det(f_{1}, M_{j_{1}}, …, M̂_{j_{n}}, …, M_{j_{g+1}})
= (−1)^{j_{n}−1}det(M_{j_{1}}, …, M_{j_{n−1}}, f_{1}, M_{j_{n+1}}, …, M_{j_{g+1}})
= (−1)^{j_{n}−1}(−1)^{k+1}t · det(N^{1k}),

where N^{1k} is the (q − 1) × (q − 1) submatrix of N obtained by deleting the first row and k-th column. Since 1 ∉ T, we have t_{T,1j} = (−1)^{j_{n}−1} and thus p_{T +1j} = t(−1)^{k+1}det(N^{1k}).

Substituting these expressions for p_{S+1j} and p_{T +1j} we get:

1 · t · det(N) + ∑_{j∈{a_{1},...,a_{q}}} t(−1)^{k}(M_{j})_{1}det(N^{1k})

which is zero in virue of a well known formula for the determinant. QED

4 The Heisenberg group and the Spin representation

4.1

We recall the basic facts on the Heisenberg group and we discuss the projective representation of its automorphism group. In Subsection 4.3 we relate the Heisenberg group (in its irreducible 2^{g}-dimensional representation) with the (half) spin representation of the orthogonal group. Combining this with previous results, we can finally write down Hitchin’s connection in Subsection 4.4.

4.2 The Heisenberg group

4.2.1 Definitions

We introduce a variant of the Heisenberg group (cf. Subsection 2.5.2). For any positive integer g we define a (finite) Heisenberg group G by:

G = G_{g} := {(t, x) : t ∈ C, t^{4} = 1, x = (ξ, ξ′) ∈ F_{2}^{g} × F_{2}^{g}}

with identity element (1, 0) and multiplication law:

(t, (ξ, ξ′))(s, (η, η′)) := (ts(−1)^{ξη′}, ξ + η, ξ′ + η′), (t, x)^{−1} = ((−1)^{ξξ′}t^{−1}, x)

with ξη′ := ∑_{i=1}^{g} ξ_{i}η′_{i} and x = (ξ, ξ′). The group G has 2^{2g+2} elements, it is non-abelian, in fact its center is {(t, 0)}. We define a symplectic form on F_{2}^{2g} by:

E(x, y) = ξη′ + ηξ′, then (t, x)(s, y)(t, x)^{−1}(s, y)^{−1} = ((−1)^{E(x,y)}, 0)

where x = (ξ, ξ′), y = (η, η′).

4.2.2 Representations.

Let V be the 2^{g} dimensional vector space of complex valued functions F_{2}^{g} → C. It has a standard basis consisting of δ-functions

X_{σ} : F_{2}^{g} ⟶ C, X_{σ}(σ) = 1, X_{σ}(ρ) = 0 if σ ≠ ρ.

The Heisenberg group G has a representation U on V, the Schrödinger representation:

(U(t, (ξ, ξ′))f)(σ) := t(−1)^{σξ′}f(σ + ξ), thus U(t, (ξ, ξ′))X_{σ} = t(−1)^{(σ+ξ)ξ′}X_{σ+ξ}.

For every x ∈ F_{2}^{2g} − {0} choose a (t_{x}, x) ∈ G of order two. We define:

U_{x} := U(t_{x}, x) (∈ GL(V)), so U_{x}^{2} = I

and define U_{0} = I. Then Im(U) = {tU_{x} : t^{4} = 1, x ∈ F_{2}^{2g}}.

4.2.3 Automorphisms.

We define a subgroup of Aut(G) by:

A(G) := {ϕ ∈ Aut(G) : ϕ((t, 0)) = (t, 0) ∀t} .

For ϕ ∈ A(G) and (t, x) ∈ G we can then write:

ϕ(t, x) := (f_{ϕ}(x)t, M_{ϕ}(x)), with M_{ϕ} : F_{2}^{2g} ⟶ F_{2}^{2g}, f_{ϕ} : F_{2}^{2g} ⟶ C^{∗}

(note ϕ(t, x) = ϕ(t, 0)ϕ(1, x) = (t, 0)ϕ(1, x)). The map M : A(G) → Aut(F_{2}^{2g}), ϕ ↦ M_{ϕ} is a homomorphism, its image is Sp(F_{2}^{2g}, E) and ker(M) ≅ G/Center(G) ≅ F_{2}^{2g} consists of interior automorfisms.

0 ⟶ F_{2}^{2g} ⟶ A(G) ⟶^{M} Sp(2g, F_{2}) ⟶ 0.

4.2.4 A projective representation.

The Schrödinger representation U of G on V is the unique irreducible representation of G in which (t, 0) acts by mutiplication by t. Given ϕ ∈ A(G), the representation U ∘ ϕ enjoys the same property. Hence by Schur’s lemma we get a linear map, unique up to scalar multiple,

T̃_{ϕ} : V ⟶ V, with T̃_{ϕ}U(h) = U(ϕ(h))T̃_{ϕ},

for all h ∈ G. Note T̃_{ϕ} normalizes Im(U). In this way we get a projective representation of A(G):

T̃ : A(G) ⟶ P GL(V), ϕ ⟼ T̃_{ϕ}.

When ϕ = Int_{(t,y)} we may take T̃_{ϕ} = U_{y}.

On the other hand, since Im(U) ≅ G any T ∈ GL(V) which normalizes Im(U) defines an automorphism ϕ_{T} of G. Since the center of G acts by scalar multiples of the identity, we have ϕ_{T} ∈ A(G). Thus we get an exact sequence

0 ⟶ C^{∗} ⟶ Normalizer_{GL(V)}(Im(U)) ⟶ A(G) ⟶ 0.

4.2.5 Example

For x = (ξ, ξ′) ∈ F_{2}^{2g} − {0}, we define the transvection

T_{x} : F_{2}^{2g} ⟶ F_{2}^{2g}, y ⟼ y + E(y, x)x.

Then T_{x} ∈ Sp(2g, F_{2}) and the (finite) symplectic group Sp(2g, F_{2}) is generated by transvections. Note that the transvections are involutions: T_{x}^{2} = 1.

Let x = (0, ξ′) with ξ′ = (1, 0, …, 0) ∈ F_{2}^{g}. On the basis X_{(0,...,0)}, …, X_{(0,τ)}, …, X_{(1,0,...,0)}, …, X_{(1,τ)}, … with τ ∈ F_{2}^{g−1} we have:

U_{x} = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) , let T̃_{x} := ( \begin{matrix}I & 0 \\ 0 & iI \end{matrix} ) .

Then it easy to verify that T̃_{x}U_{y}T̃_{x}^{−1} = t_{x,y}U_{T_{x}(y)} for all y ∈ F_{2}^{2g} and some t_{x,y} ∈ C^{∗}. Thus T̃_{x} lies in A(G) and M(T̃_{x}) = T_{x}.

4.3 The Spin representation

4.3.1

The two half spin representations of so(2g + 2) are each realized on a 2^{g}-dimensional vector space. We recall, using the Clifford algebra, how they can be constructed using the Heisenberg group. We will have to consider the Heisenberg group G_{g+1} which acts on a 2^{g+1}-dimensional vector space. Elements of order two in the Heisenberg group will define the spin representation of so(2g + 2). Restriction to suitable subspaces will give the half spin representations and their relation with G_{g} (a subquotient of G_{g+1}). This relation between so(2g + 2) and the Heisenberg group is used in 4.4.3 to prove the Heisenberg invariance of the flat connections introduced in 3.2.4.

4.3.2

In dealing with hyperelliptic curves and the half-spin representation of the orthogonal group, the following (classical) notation for points in F_{2}^{2g} is convenient ([DO], VIII.3; [M]). Let

B := {1, 2, …, 2g + 2}, then F_{B} := {f : B → F_{2} : ∑_{b∈B} f(b) = 0} /{0, [b ↦ 1]_{b∈B}}.

is an F_{2}-vector space of dimension 2g. For a subset T ⊂ B with an even number elements we denote by x_{T} ∈ F_{B} (or simply T) the element defined by the function f with f(b) = 1 iff b ∈ T. Note that x_{T} = x_{T′} when T′ is the complement of T in B, moreover

x_{T} + x_{S} = x_{R} with R = T + S := (T ∪ S) − (T ∩ S).

4.3.3

We fix the following isomorphism of F_{2}-vector spaces, and identify them in this way in the remainder of the paper ([DO], VIII.3, Lemma 2, but note we interchanged (e_{i}, 0) ↔ (0, e_{i})):

F_{B} ⟶^{≅} F_{2}^{g} × F_{2}^{g}, x_{{2i−1, 2i}} = (0, e_{i}), x_{{2i, 2i+1,...,2g+1}} = (e_{i}, 0) (1 ≤ i ≤ g).

The symplectic form E on F_{2}^{g} × F_{2}^{g} can be now be easily computed on F_{B} by:

E(x_{T} , x_{S}) = |T ∩ S| mod 2.

4.3.4

To accomodate both G_{g} and G_{g+1} we extend the construction of §4.3.2. The inclusion

B := {1, …, 2g + 2} ↪ B^{♯} := {1, …, 2g + 4}

and extension by zero of functions on B to functions on B^{♯} induces

F̃_{B} := {f : B → F_{2} : ∑_{b∈B} f(b) = 0} ↪ F̃_{B^{♯}} := {f : B → F_{2} : ∑_{b′∈B^{♯}} f(b′) = 0} .

The function g′ ∈ F̃_{B^{♯}} with g′(b′) = 1 (all b′ ∈ B^{♯}) does not lie in F̃_{B}, thus

F̃_{B} ↪ F_{B^{♯}} := F̃_{B^{♯}}/{0, g′}.

The image in F_{B^{♯}} of the function g (∈ F̃_{B}) with g(b) = 1 (all b ∈ B) is the element

p′ := x_{{1,2,...,2g+2}} = x_{{2g+3,2g+4}} ∈ F_{B^{♯}}.

Using the definition of the symplectic form on F_{B^{♯}} (§4.3.3), which we denote by E^{♯}, one finds that:

F̃_{B} = (p′)^{⊥}, with p^{′⊥} := {x ∈ F_{B^{♯}} : E^{♯}(x, p′) = 0 } and F_{B} = (p′)^{⊥}/{0, p′}.

From §4.3.3 we have an identification:

F_{B^{♯}} = F^{g+1} × F^{g+1}, p′ = (0, (1, 1, …, 1)) (∈ F_{B^{♯}}).

The Heisenberg group defined by this identification (cf. §4.2.1) will be denoted by G^{♯}, its Schrödinger representation by U^{♯} (on the 2^{g+1}-dimensional vector space V^{♯} := {f : F_{2}^{g+1} → C} with basis of δ-functions Y_{σ′}, σ′ ∈ F_{2}^{g+1}).

For any x′ ∈ (p′)^{⊥} (⊂ F_{B^{♯}}), the maps U_{p′}^{♯} and U_{x′}^{♯} commute. Therefore the U_{x′}^{♯} with x′ ∈ (p′)^{⊥} act on the two eigenspaces of U_{p′}^{♯} which are:

V_{+}^{♯} := ⟨…, Y_{σ′}, …⟩_{σ′_{1}+...+σ′_{g+1}=0}, V_{−}^{♯} := ⟨…, Y_{σ′}, …⟩_{σ′_{1}+...+σ′_{g+1}=1}.

A quotient map from (p′)^{⊥} (⊂ (F_{2}^{g} × F_{2}) × (F_{2}^{g} × F_{2})) to F_{B} = F_{2}^{g} × F_{2}^{g} with kernel {0, p′} is:

(p′)^{⊥} ⟶ F_{B} = (p′)^{⊥}/{0, p′} ≅ F_{2}^{2g}, x′ := ((a, a_{g+1}), (b, b_{g+1})) ⟼ x := (a, b̄)

with b̄ = (b_{1} + b_{g+1}, …, b_{g} + b_{g+1}). We define an isomorphism of vector spaces:

V_{+}^{♯} ⟶ V, Y_{σ_{1},...,σ_{g},σ_{g+1}} ⟼ X_{σ_{1},...,σ_{g}}.

4.3.5 Lemma. For any x′ ∈ (p′)^{⊥} mapping to x ∈ F_{B} there is a commutative diagram:

V_{+}^{♯} ⟶^{≅} V
U^{♯}(t, x′) ↓  ↓ U(t, x)
V_{+}^{♯} ⟶^{≅} V

Moreover, if 1 ≤ j, k ≤ 2g + 1, then restriction of U_{x_{jk}}^{♯} to V_{+}^{♯} ≅ V is given by ±U_{x_{ij}}.

Proof. The proof consists of easy computations which we leave to the reader. QED

4.3.6 The Clifford algebra

The Clifford algebra C(Q) of the quadratic form Q = x_{1}^{2} + … + x_{2g+2}^{2} on V_{Q} = C^{2g+2} is the quotient of the tensor algebra T (V_{Q}) by the two-sided ideal I generated by the elements v ⊗ v − Q(v) for v ∈ V_{Q} ([FH], §20.1):

C(Q) := T (V_{Q})/I = (C ⊕ V_{Q} ⊕ V_{Q}⊗V_{Q} ⊕ …)/(…, −Q(v) + v ⊗ v, …)_{v∈V_{Q}}.

Let e_{1}, …, e_{2g+2} be the standard basis vectors of V_{Q}, then the C(Q) is generated by C and the e_{j} with relations:

e_{j} · e_{j} = 1, e_{j} · e_{k} + e_{k} · e_{j} = 0 (j ≠ k),

here · stands for the product induced by ⊗ on C(Q). The (associative) algebra C(Q) becomes a Lie algebra by defining, as usual, the Lie bracket to be [x, y] := x · y − y · x.

The following proposition relates the Heisenberg group, the Clifford algebra and the spin representation of the Lie algebra so(2g + 2).

4.3.7 Proposition.

With the notation as above we have:

1. The C-linear map

γ^{♯} : C(Q) ⟶ End(V^{♯}),  λ ↦ λI, e_{k} ↦ U_{x_{k, 2g+4}}^{♯}  (λ ∈ C, k = 1, …, 2g + 2)

defines an isomorphism of (associative C-) algebras.

2. The linear map:

ρ_{s}^{♯} : so(2g + 2) ⟶ End(V^{♯}) ≅_{γ^{♯}} C(Q), F_{jk} ⟼ U_{x_{j,2g+4}}^{♯}U_{x_{k,2g+4}}^{♯} (= γ^{♯}(e_{j} · e_{k}))

is an injective homomorphism of Lie algebras.

3. The subspace V_{+}^{♯} of V^{♯} is invariant under the action of so(2g + 2). The Lie algebra representation

ρ_{s} : so(2g + 2) ⟶ End(V_{+}^{♯}) ≅ End(V), ρ_{s}(x) := ρ_{s}^{♯}(x) |_{V_{+}^{♯}}

is an (irreducible) half spin representation of so(2g + 2). In particular, V ≅ V (ω_{g+1}).

4. We have

ρ_{s}(F_{jk}) = ±iU_{x_{jk}}, (F_{jk} ∈ so(2g + 2), i^{2} = −1),

where the U_{x_{jk}} (∈ End(V)) with x_{jk} ∈ F_{2}^{2g} are in the Schrödinger representation of the Heisenberg group G.

Proof. By definition we have (U_{x′}^{♯})^{2} = I for all x′ ∈ F_{2}^{2g+2}. Moreover, E^{♯}(x_{{k,2g+4}}, x_{{l,2g+4}}) = 1 when k ≠ l so the corresponding maps anti-commute: U_{x_{k,2g+4}}^{♯}U_{x_{l,2g+4}}^{♯} = −U_{x_{l,2g+4}}^{♯}U_{x_{k,2g+4}}^{♯}. Then the map γ^{♯} preserves the relations in C(Q) and is thus an algebra homomorphism. It is surjective because the matrices U_{x′}^{♯}, where x′ runs over F_{2}^{g+1} ×F_{2}^{g+1} are a basis of End(V^{♯}) (the Schrödinger representation being irreducible) and any U_{x′}^{♯} is a product (up to scalar multiple) of suitable U_{x_{j,2g+4}}^{♯}. Therefore γ^{♯} is an isomorphism since both algebras have the same dimension.

(2) This is worked out in [FH], Lemma 20.7. With their notations and our Q, we have

ϕ : ∧^{2}V_{Q} ⟶^{≅} so(2g + 2), e_{j} ∧ e_{k} ⟼ F_{jk},

ψ : ∧^{2}V_{Q} ⟶ C(Q),  e_{j} ∧ e_{k} ⟼ e_{j} · e_{k}  (j ≠ k),  and ρ_{s}^{♯} = ψ ∘ ϕ^{−1}.

(3) By definition of ρ_{s}^{♯} and the fact that U^{♯} is a representation we have:

ρ_{s}^{♯}(F_{jk}) = U_{x_{j,2g+4}}^{♯}U_{x_{k,2g+4}}^{♯} = c_{jk}U_{x_{jk}}^{♯},

with 1 ≤ j, k ≤ 2g + 2 and some c_{jk} ∈ C^{∗}. Thus the ρ_{s}^{♯}(F_{jk}) commute with U_{p′}^{♯} in End(V^{♯}). Therefore we obtain a Lie algebra representation on V_{+}^{♯} cf. [FH], Prop. 20.15 (and identify their End(∧^{even}W) with our End(V_{+}^{♯})), where also the irreducibility is proved.

(4) We have ρ_{s}^{♯}(F_{jk}) = U_{x_{j,2g+4}}^{♯}U_{x_{k,2g+4}}^{♯}, thus ρ_{s}^{♯}(F_{jk})^{2} = −I since these two elements anticommute. Therefore ρ_{s}^{♯}(F_{jk}) = ±iU_{x_{jk}}^{♯}, which acts as ±iU_{x_{jk}} on V_{+}^{♯}. QED

4.4 The Hitchin connection

4.4.1

The symmetric group S_{2g+2} acts on P by permuting the coordinates, the quotient wil be denoted by \overline{P}. The fundamental group of P is the pure braid group and π_{1}(\overline{P}) = B_{2g+2}, the Braid group.

In case g = 2, we have S_{6} ≅ Sp(4, F_{2}), in fact there is a surjective map (which factors over the mapping class group)

π_{1}(\overline{P}) = B_{6} ⟶ Sp(4, Z),

the kernel of the composition B_{6} → Sp(4, Z) → Sp(4, F_{2}) is π_{1}(P).

From the theory of theta functions we know that the groupscheme G is trivialized on the cover P̃ of P defined by the kernel of the composition:

ϕ : π_{1}(\overline{P}) ⟶ Sp(4, Z) ⟶ Sp(4, Z)/Γ_{2}(2, 4) ≅ A(G).

Here we use Igusa’s notation:

Γ_{2}(2, 4) := \{( \begin{matrix}I + 2A & 2B \\ 2C & I + 2D \end{matrix} ) ∈ Sp(4, Z) : diagonal(B) ≡ diagonal(C) ≡ (0, 0) mod 2\} .

Thus we consider the following diagram of etale Galois coverings:

P̃ ⟶ P ⟶ \overline{P} := P/S_{6},

and the corresponding exact sequence of covering groups:

0 ⟶ F_{2}^{2g} ⟶ A(G) ⟶ S_{6} ⟶ 0.

Recall that we have defined a projective representation T̃ : A(G) → P GL(V) in 4.2.4. This induces projective representations on each S^{k}V.

4.4.2 Theorem. With the notation as in 4.4.1 (except for S_{k}) and g = 2 we have:

1. There is a line bundle N on P̃ such that the pull-back of p_{∗}L^{⊗k} is isomorphic to S_{k} ⊗_{C} N. Here S_{k} = S^{k}V (ω_{g+1}) and V (ω_{g+1}) is a half spin representation of so(6) ≅ sl(4) (which is the standard representation (or its dual) of sl(4)).

2. The Hitchin connection on the pull-back of p_{∗}L^{⊗k} to P̃ is given by the pull-back to P̃ of the connection on S_{k} ⊗_{C} O_{P} defined by the one form

\frac{−1}{16(k + 2)} ∑_{i,j i≠j} \frac{σρ̃_{s}(Ω_{ij})dz_{i}}{z_{i} − z_{j}}

with ρ̃_{s} : U(so(2g + 2)) → End(S_{1}) the half spin representation (then the σρ̃_{s}(Ω_{ij}) give endomorphisms of each S_{k}, cf. 3.2.5).

3. The Hitchin connection on the bundle S_{k} ⊗ O_{P} over P̃ descends to a projective flat connection on the the bundle (S_{k} ⊗ O_{P})/A(G) over \overline{P}, where A(G) acts on S_{k} via the k-th symmetric power of its projective representation on S_{1} and A(G) acts on P̃ as Gal(P̃/\overline{P}). This descent gives the kth Hitchin connection of the natural descend of the family of curves described in 1.4.

Proof. For the first part it suffices to show that the pull-back of p : P → P to P̃ is trivial. Since G is isomorphic to the constant group scheme G (cf. 4.2.1 (with here t ∈ C^{∗})), the uniqueness of the Schrödinger representation gives the global triviallization. We already observed that the line bundle N does not interfere with the projective connections.

(2) We use Theorem 2.6.9 so we must verify (Eq.) from 2.6.7. The covariant derivatives given are defined by the heat operator

X = −\frac{1}{16(k + 2)} ∑_{j≠i} σρ̃_{s}(Ω_{ij}) ⊗ \frac{dz_{i}}{z_{i} − z_{j}} (∈ (S_{2} ⊗_{C} S_{2}^{∗}) ⊗ Ω_{P}^{1}).

The Heisenberg-invariance of this operator follows from the facts that Ω_{ij} = F_{ij} ⊗ F_{ij}, that ρ_{s}(F_{kj}) = ±iU_{x_{jk}} with i^{2} = −1 and that U_{y}U_{x}U_{y}^{−1} = ±U_{x}. Since S^{4}(ρ̃_{s}) is a subrepresentation of ρ̃: so(2g + 2) → EndS^{2}(∧^{2}C^{2g+2}) we have σρ̃_{s}(Ω_{ij}) = σρ̃(Ω_{ij}) on S_{4}. For the connection ∇_{0} on p̃_{∗}O_{P} (1) = S_{1} ⊗ OP̃ we simply take ∇_{0}(fw) = w ⊗ df. Then the equation (Eq.) is exactly the statement of Theorem 3.4.3.

(3) We have a smooth projective family of genus 2 curves over the scheme P̄ given by

y^{2} = f(x) = x^{6} + a_{1}x^{5} + a^{2}x^{4} + ... + a_{6} = ∏_{i}(x − z_{i}),

where a_{i} is the i-th symmetric function of the z_{j}. Thus by the general theory of Section 2 we have a P^{3}-bundle p̄ : P̄ ⟶ P̄ and a projective Hitchin connection D̄_{k} on p̄_{∗}O(k). Let φ : P̃ ⟶ P̄ be the natural map. Then there is an isomorphism

(∗) φ^{∗}(P̄) = P^{3} × P̃

which is compatible with the action of A(G) (which lies over the action of A(G) on P̃) on both sides: this isomorphism is given by the trivialization of the action of the theta-group scheme G which was mentioned earlier. The isomorphism (*) induces an isomorphism

(∗∗) φ^{∗}(p̄_{∗}O(k)) ≅ S_{k} ⊗ OP̃

which is compatible with the natural projective action of A(G) on the LHS (via G) and the action of A(G) on the RHS via the Sym^{k} of its natural representation on S_{1}. The Heat operator given by X (as in proof of 4.4.2.2, see 2.6.7) on the LHS of (*) is VIA (*) compatible with the pullback of the Hitchin heat operator of the LHS of (*). But both heat operators are invariant under the action of A(G). Thus the projective connections on the RHS and LHS of (**) agree and are compatible with the projective action of A(G) on both sides. In other words the natural descent datum on the LHS of (**) agrees with the descent datum on the RHS of (**). QED

4.4.3 Examples

We give some examples of the ρ_{s}(Ω_{ij})’s in case g = 2. We identify V = S_{1}, with its standard basis X_{σ}, σ ∈ F_{2}^{2}. A linear map with matrix (a_{kl}) is then also given by the linear differential operator ∑_{kl} a_{kl}X_{k}∂_{l} (since ∂_{l}(X_{m}) = 0 for l ≠ m and is 1 if l = m). Recall from 4.3.3 that (for g = 2):

x_{12} = ((0, 0), (1, 0)), so U_{x_{12}} = X_{00}∂_{00} + X_{01}∂_{01} − X_{10}∂_{10} − X_{11}∂_{11}.

Similarly, we had x_{16} = ((1, 0), (0, 0)) and x_{26} = ((1, 0), (1, 0)) and so:

U_{x_{16}} = X_{10}∂_{00} + X_{11}∂_{10} + X_{00}∂_{10} + X_{01}∂_{10}, U_{x_{26}} = i(−X_{10}∂_{00} − X_{11}∂_{10} + X_{00}∂_{10} + X_{01}∂_{10}).

Since U_{x_{jk}} = ±iρ_{s}(F_{jk}) and U_{x_{jk}}^{2} = I we get (cf. 3.2.5):

σ(ρ̃_{s}(Ω_{12})) = −2(X_{00}^{2}∂_{00}^{2} + X_{01}^{2}∂_{01}^{2} + X_{10}^{2}∂_{10}^{2} + X_{11}^{2}∂_{11}^{2} + 2X_{00}X_{01}∂_{00}∂_{01} − 2X_{10}X_{11}∂_{10}∂_{11}),

4.5 Local Monodromy

4.5.1 Introduction

We want to obtain some information on the representation of the mapping class group for genus two curves defined by Hitchin’s connection. This representation has been studied, for arbitrary g and k, by Moore and Seiberg [MS], Kohno [K2] and in the case k = 2 by Wright [Wr]. In some sense, we merely find (weaker, local) results which do agree with their results.

4.5.2 The method

We recall how to determine the local monodromy on the trival bundle O_{P} ⊗_{C} W with a connection ∇. We consider a holomorphic map

ϕ : D^{∗} ⟶ P, with D^{∗} := {t ∈ C − {0} : |t| < ϵ }.

(for small, positive ϵ) and pull-back the one form ω ∈ End(W)⊗Ω_{P}^{1} which defines the connection (so ∇(fw) = wdf − fω(w). We write

ϕ^{∗}ω = (\frac{R}{t} + A(t)) dt, R ∈ End(W)

with A(t) holomorphic for t = 0, and R is called the residue at t = 0. We identify π_{1}(D^{∗}) = Z; 1 ∈ Z represents a small circle traversed in anti-clockwise direction. The eigenvalues of the monodromy of 1 ∈ π_{1}(D^{∗}) are the exp(−2πiμ_{j}) were the μ_{i} are the eigenvalues of R on W.

In case the monodromy transformations are semi-simple, the local monodromy is conjugated to exp(−2πiR) (cf. [D], p. 54). However we could not prove the semi-simplicity (but it seems to be known to the physicists).

To get the eigenvalues of the monodromy of the Hitchin connection for a γ ∈ π_{1}(\overline{P}), let n be the order of its image in Gal(P/\overline{P}) and let γ̄ be its image in A(G) = Gal(P̃/\overline{P}). We choose a ϕ : D^{∗} → P in such a way that ϕ_{∗}(1) = γ^{n} (∈ π_{1}(P)). Then these eigenvalues are the eigenvalues of the matrix exp(\frac{−2πi}{n}R)T̃γ̄, with R determined as above for the corresponding connection on P (cf. [K1]). Since T̃ is a projective representation (on S_{1} and thus on any S_{k}), the set of eigenvalues are only defined up to multiplication by one non-zero constant. This corresponds to the fact that we only have a projectively flat connection.

4.5.3 Non-separating vanishing cycle

Let γ ∈ π_{1}(P̄) such that γ^{2} = ϕ_{∗}(1) with

ϕ : D^{∗} ⟶ P, t ⟼ (t + z_{2}, z_{2}, …, z_{6})

where we fix distinct z_{j} ∈ C. (Then γ corresponds to a Dehn twist in a simple nonseparating loop in the mapping class group.) The residue of ϕ^{∗}ω for the connection on S_{k} ⊗ O_{P} corresponding to the Hitchin connection on p_{∗}L^{⊗k} is (cf. 4.4.2 and 4.4.3): \frac{−1}{16(k+2)}σρ̃_{s}(Ω_{12}),

−ρ̃_{s}(Ω_{12}) = R_{12} := ((X_{00}∂_{00} + X_{01}∂_{01}) − (X_{10}∂_{10} + X_{11}∂_{11}))^{2} .

4.5.4 Lemma. The eigenvalues of of the monodromy of γ as in 4.5.3 on S_{k} are:

λ_{c} := exp(−2πi\frac{c(c+1)}{k+2}), mult(λ_{c}) = (k − 2c + 1)(2c + 1)

(up to multiplication by a non-zero constant independent of c), with 2c ∈ Z and 0 ≤ c ≤ \frac{k}{2}.

Proof. Since the eigenvalues are only determined up to a constant and since the difference between 2R_{12} and σ(R_{12}) is a multiple of the Euler vector field, which acts by multiplication by k on S_{k}, it suffices to consider the eigenvalues of R_{12}. For a monomial in S_{k} we have:

R_{12}(X_{00}^{l_{0}}X_{01}^{l_{1}}X_{10}^{m_{0}}X_{11}^{m_{1}}) == ((l_{0} + l_{1}) − (m_{0} + m_{1}))^{2}(X_{00}^{l_{0}}X_{01}^{l_{1}}X_{10}^{m_{0}}X_{11}^{m_{1}})

Thus each monomial is an eigenvector. Let b := m_{0} + m_{1}. Since the monomial has degree k, its eigenvalue is (k − 2b)^{2}. The multiplicity of the eigenvalue is (k − b + 1)(b + 1) (the dimension of the space of homogeneous polynomials in 2 variables of degree c is c + 1). Thus the eigenvalue 2(k − 2b)^{2} of 2R_{12} has multiplicity (k − b + 1)(b + 1).

The image γ̄ of γ in A(G) is T̃_{x_{12}} which acts on such a monomial by (cf. 4.2.5):

T̃_{x_{12}}(X_{00}^{l_{0}}X_{01}^{l_{1}}X_{10}^{m_{0}}X_{11}^{m_{1}}) = i^{b}(X_{00}^{l_{0}}X_{01}^{l_{1}}X_{10}^{m_{0}}X_{11}^{m_{1}}).

The eigenvalues of γ on S_{k} are then, with c := b/2:

\begin{aligned}exp(\frac{−2πi}{2} (\frac{+1}{16(k+2)}2(k − 2b)^{2} + \frac{b}{4})) & = & exp(−2πi (\frac{k^{2}−8ck+16c^{2}+8ck+16c}{16(k+2)})) \\ & = & exp(−2πi (\frac{k^{2}}{16(k+2)})) exp(−2πi (\frac{c(c+1)}{k+2})). \end{aligned}

QED

4.5.5 Separating vanishing cycle

Now we consider a γ ∈ π_{1}(P̄) such that γ = ϕ_{∗}(1) with

ϕ : D^{∗} ⟶ P, t ⟼ (tz_{1}, tz_{2}, tz_{3}, z_{4}, z_{5}, z_{6}),

with distinct nonzero z_{i}’s.

The residue in t = 0 of the connection on S_{k} ⊗ P corresponding to Hitchin’s connection is

\frac{−1}{16(k+2)}R_{123}, with R_{123} := 2σ(ρ̃_{s}(Ω_{12} + Ω_{13} + Ω_{23})).

4.5.6 Lemma. Let g = 2.

1. There are constants λ_{k} ∈ Q such that (in End(S_{k})):

R_{123} = 16R_{Q} + λ_{k}I with R_{Q} = (X_{00}X_{10} − X_{01}X_{11})(∂_{00}∂_{10} − ∂_{01}∂_{11}).

2. We define a subspace of S_{k} by:

V_{k} := Kernel(X_{Q} := ∂_{00}∂_{10} − ∂_{01}∂_{11} : S_{k} ⟶ S_{k−2}).

Then the vector space S_{k} is a direct sum:

S_{k} = V_{k} ⊕ QV_{k−2} ⊕ Q^{2}V_{k−4} ⊕ …

Moreover, each subspace Q^{l}V_{k−2l} is an eigenspace of R_{Q} with eigenvalue:

λ_{l} = l(k − l + 1) and dim V_{k−2l} = (k − 2l + 1)^{2}.

3. The eigenvalues of of the monodromy of γ on S_{k} are

λ_{l} := exp(−2πi\frac{l(l+1)}{k+2}) mult(λ_{l}) = (k − 2l + 1)^{2}

(up to multiplication by a constant independent of l), with l ∈ Z and 0 ≤ l ≤ k/2.

Proof. Recall x_{12} corresponds to ((0, 0), (1, 0)) ∈ F_{2}^{2}×F_{2}^{2} and x_{23} = x_{{2,3,4,5}}+x_{{4,5}} corresponds to ((1, 1), (0, 0)) thus x_{13} = x_{12} + x_{23} corresponds to ((1, 1), (1, 0)). Since (1, 1)(1, 0) = 1, we have a + sign in front of Ω_{13} below. Let ρ̃_{s}(Ω_{00}) be the square of the Euler vector field (which acts as k^{2}I on S_{k}). Then:

\begin{aligned}ρ̃_{s}(Ω_{00}) & = & +(X_{00}∂_{00} + X_{01}∂_{01} + X_{10}∂_{10} + X_{11}∂_{11})^{2}, \\ ρ̃_{s}(Ω_{12}) & = & −(X_{00}∂_{00} + X_{01}∂_{01} − X_{10}∂_{10} − X_{11}∂_{11})^{2}, \\ ρ̃_{s}(Ω_{23}) & = & −(X_{11}∂_{00} + X_{10}∂_{01} + X_{01}∂_{10} + X_{00}∂_{11})^{2}, \\ ρ̃_{s}(Ω_{13}) & = & +(X_{11}∂_{00} + X_{10}∂_{01} − X_{01}∂_{10} − X_{00}∂_{11})^{2}. \end{aligned}

The sum of these operators, minus the (degree one) parts which act as constants on each S_{k}, is 4R_{Q}. Remembering the factor 2 in R_{123} and in our symbols, we find the first statement.

(2) This is well-known. Let f ∈ V_{n}, so f is homogeneous of degree n and X_{Q}(f) = 0. Then an elementary computation gives for l ≥ 1 (with variables X_{i} and Q = X_{0}X_{1} − X_{2}X_{3}):

(∂_{0}∂_{1} − ∂_{2}∂_{3})(fQ^{l})
= (∂_{0}f)(∂_{1}Q^{l}) + (∂_{0}Q^{l})(∂_{1}f) − (∂_{2}f)(∂_{3}Q^{l}) − (∂_{2}Q^{l})(∂_{3}f) + f(∂_{0}∂_{1} − ∂_{2}∂_{3})(Q^{l})
= lQ^{l−1}(X_{0}∂_{0} + X_{1}∂_{1} + X_{2}∂_{2} + X_{3}∂_{3})(f) + l(l + 1)fQ^{l−1}
= l(n + l + 1)fQ^{l−1}.

Writing n = k − 2l we get, for all integers l with 0 ≤ 2l ≤ k:

R_{Q}(fQ^{l}) = l(k − l + 1))fQ^{l}, f ∈ V_{k−2l}.

Hence each V_{k−2l}Q^{l} is an eigenspace for R_{Q}. By induction (the cases k = 0, 1 being trivial) we may assume that S_{k−2} = ⊕S_{k−2−2l}Q^{l}. Therefore QS_{k−2} = ⊕Q^{l+1}S_{k−2(l+1)} ⊂ S_{k} is a direct sum of eigenspaces of R_{Q} and none of the eigenvalues of R_{Q} on this subspace is zero (k + l(l + 1) > 0 for k ≥ 2, l ≥ 0). Thus ker(R_{Q}) ∩ QS_{k−2} = {0} and R_{Q} induces an isomorphism on QS_{k−2}. Therefore S_{k} = Ker(R_{Q}) ⊕ Im(R_{Q}) = ⊕Q^{l}S_{k−2l}. The dimension of V_{k} is then dim S_{k} − dim S_{k−2} = \binom{k+3}{3} − \binom{k+1}{3} = (k + 1)^{2}.

(3) The image of γ in A(G) is trivial. The eigenvalues of R_{123} = 16R_{Q} on S_{k} are 16l(k−l+1). Thus the eigenvalues of γ are (see 4.5.2):

\begin{aligned}exp(−2πi\frac{−1}{16(k+2)}16l(k − l + 1)) & = & exp(−2πi (\frac{l^{2}−l−l(k+2)+2l}{k+2})) \\ & = & exp(−2πi\frac{l(l+1)}{k+2}). \end{aligned}

QED 4.5.7 Kohno’s results

The monodromy of the Hitchin connection has been studied by Moore-Seiberg and Kohno (among others), we will relate the results above to those contained in [K2] for g = 2.

Let γ be a tri-valent graph with two vertices, so there are three edges meeting in each vertex. For k ∈ Z_{≥0} we define a finite set b_{k}(γ) of functions on the edges of γ by:

f : Edges(γ) ⟶ \{0, \frac{1}{2}, 1, …, \frac{k}{2}\} with \{ \begin{matrix}f(e_{i}) + f(e_{j}) + f(e_{k}) ≤ k, \\ |f(e_{i}) − f(e_{j})| ≤ f(e_{k}) ≤ f(e_{i}) + f(e_{j}), \\ f(e_{i}) + f(e_{j}) + f(e_{k}) ∈ Z, \end{matrix}

for any three edges e_{i}, e_{j}, e_{k} meeting in a vertex. The Verlinde space V_{k} is the C-vector space with basis b_{k}(γ), it has the same dimension as S_{k}.

The graph γ corresponds to a pants decomposition of a genus two Riemann surface. Each vertex is a pair of pants, homeomorphic to P^{1} minus 3 points, the edges correspond to these points, which in turn correspond to ‘vanishing cycles’ on the Riemann surface. The mapping class group has a projective representation on V_{k}. The Dehn twist in a vanishing cycle corresponding to an edge e of γ acts, up to a scalar multiple, conjugated to the diagonal matrix with entries (cf. [K2], p. 217; p. 214, (2-1))

exp(−2πif(e)(f(e) + 1)/(k + 2) (f ∈ b_{k}(γ)).

Using this recipe we find the same results as before (see Lemmas 4.5.4 and 4.5.6):

4.5.8 Lemma. With the notation as above:

1. The eigenvalues of the Dehn twist in a non-separating cycle on V_{k} are exp(−2πiv(v + 1)/(k + 2)) with multiplicity (k − 2v + 1)(2v + 1) where v ∈ \{0, \frac{1}{2}, 1, …, \frac{k}{2}\}.

2. The eigenvalues of a Dehn twist in a separating cycle on V_{k} are exp(−2πil(l + 1)/(k + 2)) with multiplicity (k − 2l + 1)^{2} where l is an integer with 0 ≤ l ≤ k/2.

Proof. For the first statement consider the graph γ with two vertices joined by three edges e_{1}, e_{2}, e_{3}. The edge e_{1} corresponds to a non-separating cycle and one must determine f(e_{1}) for all f ∈ b_{k}(γ).

For the second case the recipe is not explicitly given in [K2], but is similar to the one used in the first case and gives a result that agrees with Lemma 4.5.6. One now takes δ to be the graph with two vertices v_{1}, v_{2} and three edges e_{1}, e_{2}, e_{3} such that begin and end of e_{i} is v_{i} and where e_{3} connects v_{1} and v_{2}. Thus the edge e_{3} corresponds to a separating vanishing cycle and one has to determine f(e_{3}) for all f ∈ b_{k}(δ). The computations are elementary and we leave them to the reader. QED 4.5.9 The cases k = 1, 2

The case k = 1 is particularly easy since the operators Ω_{ij} are homogeneous of degree two and thus act as zero on S_{1}. The monodromy representation then factors over the projective representation of A(G).

In case k = 2, the space S_{k} is a direct sum of 10 distinct, one dimensional G-representations (this is easy to verify, see also [vG]). On each eigenspace, any σρ̃_{s}(Ω_{ij}) acts as scalar multiplication by an integer. A (local) basis of flat sections of S_{2} ⊗ P is then given by 10 functions of the type (∏(z_{i} −z_{j})^{r_{ijm}})Q_{m} with r_{ijm} ∈ Q and Q_{m} ∈ S_{2} is a basis of the eigenspace. These eigenspaces are permuted by the projective representation of A(G).

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Bert van Geemen, Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy. geemen@dm.unito.it

A.J. de Jong, Princeton University, Department of Mathematics, Fine Hall – Washington Road, Princeton, NJ 08544-1000, USA. dejong@math.Princeton.EDU