THE HITCHIN CONNECTION IN ARBITRARY CHARACTERISTIC

Abstract

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan–Atiyah–Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson–Schechtman and Bloch–Esnault.

1 Introduction

1.1

The Hitchin connection was originally introduced in [30], with a twofold motivation. The first was an elucidation of the $2+1$ -dimensional topological quantum field theory proposed by Witten to explain the polynomial Jones invariants for knots [59, 6]. The second was the question of the dependency of the geometric quantisation of a symplectic manifold on the choice of polarisation.

In a beautiful construction, Hitchin exhibited a flat projective connection on the bundles of nonabelian theta functions over the base of a family of compact Riemann surfaces. For a fixed Riemann surface, the corresponding vector space can be understood to be the geometric quantisation of the moduli space of flat unitary connections on the underlying surface. The latter carries a canonical symplectic structure, but the complex structure on the surface also equips the moduli space with a Kähler polarisation, and the connection indicates precisely how the quantisation varies.

Even though the construction of the connection uses analytic and Kähler techniques throughout, it was already observed by Hitchin that the end result could entirely be interpreted in terms of algebraic geometry and should in fact hold in positive characteristic as well (see [31, §5]). This in itself is not too surprising, bearing in mind that one of the sources of inspiration for Hitchin was the work of Welters [58], which generalised the heat equation that (abelian) theta functions had classically been know to satisfy to positive characteristic. Welters work was probably the first in which a cohomological approach to heat equations was developed; the nonabelian situation is quite a bit more involved, however.

The aim of this paper now is to give a new, purely algebro-geometric, construction of the Hitchin connection, without using any analytic or Kähler techniques. This construction works as well in positive characteristic (apart from a few exceptions, see below), which as far as we are aware is a first for either the Hitchin connection itself or any of the equivalent connections (such as the KZB or TUY/WZW connection from conformal field theory – see, however, [51] for a recent study of the KZ equation in positive characteristic). We stress that the construction only involves (finite-dimensional) algebraic geometry and, in particular, no infinite-dimensional representation theory – the only prerequisites needed are covered by [27].

Key elements in our construction are a framework for connections coming from heat operators in algebraic geometry, due to van Geemen and de Jong [57], as well as a substitute for the Narasimhan–Atiyah–Bott Kähler form [42, 1], which according to Quillen [45] realizes the Chern class of the determinant-of-cohomology line bundle. The serendipitous similarity between this Kähler form and the quadratic part of the Hitchin system were crucially used in [30] to obtain the Hitchin connection in the complex case.

We compensate for the absence of this Kähler form by interpreting the cohomology class of the line bundle as an Atiyah class. This difference in guaranteeing the cohomological conditions of the theorem of van Geemen and de Jong forms the bulk of our work.

An essential ingredient of our construction is the description of the Atiyah algebra of the theta line bundle over the moduli space in terms of the first direct image of the Atiyah algebra of a universal bundle (Theorem 4.4.1). A complete proof is given in section 5 and in appendices A and B, whose aim is to give a simplified and self-contained presentation of the results used in the proof of this theorem, i.e., the theory of the trace complex [16], [9] and some additional inputs worked out in [50], describing the behaviour of the above objects when replacing a universal bundle by its endomorphism bundle. We observe that the paper [50] also describes a construction of the Hitchin connection, but the strategy in [50] is different from ours: They construct the Hitchin connection by relying on another argument from [21], whereas our approach seeks to verify directly the van Geemen–de Jong criterion for the liftability of a symbol map to a heat operator.

1.2

At this point, we would like to make a few comments on the relationship of this work to the existing literature. As already mentioned, we will follow the algebro-geometric framework of van Geemen and de Jong [57] for connections induced by heat operators. This provides a purely cohomological criterion for the existence of a heat operator with a prescribed symbol map.

In [57, §2.3.8] van Geemen and de Jong show how their framework of connections induced by heat operators easily recaptures Welters’ construction of the Mumford–Welters projective connection on bundles of theta functions. The main point of their work is to use this framework (which we resume below in Theorem 3.4.1) to construct a Hitchin connection (in complex algebraic geometry) in the particular case of rank $2$ bundles on genus $2$ curves (which was excluded from Hitchin’s original work and indeed from ours as well). They do not reestablish the Hitchin connection in all other cases though, and in this sense the present paper exactly complements their work.

We remark that several other algebro-geometric descriptions of connections on bundles of nonabelian theta functions have appeared in the literature – e.g., [21, 46, 23, 47, 50, 18]. It is not always clear, however, exactly how these connections are related (see, e.g., [4]), and for various reasons they are all restricted to characteristic zero. None also directly use the framework of van Geemen and de Jong. We remark that many of the properties of Hitchin’s original connection like, e.g., monodromy [37] or projective flatness of strange duality maps [11] have been proved with representation-theoretical methods, more precisely by using its equivalence, due to Laszlo [35], with the Tsuchiya-Ueno-Yamada (TUY)/Wess-Zumino-Witten (WZW) connection on spaces of conformal blocks [55, 54]. For most of the cited works, the relationship with conformal blocks is undeveloped (they have of course other motivations: e.g., [50], which together with [23] is probably closest to our approach, is particularly focused on the logarithmic description of the connection as the curves degenerate to nodal singularities). We therefore thought it useful to establish the Hitchin connection itself, in the original context (moduli of bundles with trivial determinant over curves), in a purely algebro-geometric way that nevertheless manifestly gives the same connection as Hitchin and to which Laszlo’s theorem immediately applies. For completeness, we mention that there are several other constructions in the literature of a differential geometric or Kähler nature, e.g., [3, 2, 49].

We want to mention that (because of Laszlo’s theorem) the term Hitchin connection is often loosely employed to refer to any of a number of equivalent projective connections. We shall use it in a much stricter sense, however, as a connection arising through a heat operator with a prescribed symbol map (see below).

In this context, the terminology nonabelian theta functions is frequently used (including by us), even though that is in fact slightly misleading. Our construction of the connection only works for moduli spaces of bundles with trivial determinant, or equivalently, $\operatorname {SL}(r)$ -principal bundles. At various places the (semi-)simplicity is crucial, and as far as we are aware, there is currently no construction that works immediately for arbitrary reductive groups. Indeed, a connection for moduli of $\operatorname {GL}(r)$ -principal bundles was crucially needed in [11], but this was created out of an $\operatorname {SL}(r)$ -connection and an (abelian) $\mathbb {G}_m$ -connection.

1.3

As a motivation for looking at the Hitchin connection from a purely algebro-geometric point of view, we would like to highlight three contexts. The first is the Grothendieck–Katz p-curvature conjecture [33], which (roughly speaking) claims that every algebraic connection which is formulated in sufficient generality and has vanishing p-curvature when reduced mod p for almost all p should have finite monodromy in the complex case. Presumably motivated by this conjecture, it was originally expected (see [13, §7]) that the Hitchin connection would have finite monodromy. However, it was shown by Masbaum in [40] that, for rank $2$ , the image of the corresponding projective representation of the mapping class group will, for all genera and almost all levels, contain elements of infinite order. This came somewhat as a surprise, as the connection for abelian theta functions was well known to have finite monodromy from Mumford’s approach through theta groups. Masbaum was working with a skein-theoretic approach to these representations, but the equivalence of this picture with the Hitchin connection follows from the work of Andersen and Ueno [7] combined with Laszlo’s theorem. Masbaum’s result was also directly rederived in an algebro-geometric context by Laszlo, Sorger, and the fourth named author [37]. We hope that our construction can be a starting point for investigating the p-curvature of the Hitchin connection.

The second is the question of integrality of topological quantum field theories (TQFTs) and the related topic of modular representations of the mapping class group. Various results have been obtained here through a skein-theoretic approach, cfr. [22, 24, 25, 26], but so far a geometric counterpart is missing. We again hope that the current work can help shed light on these issues.

Finally, we would like to mention various generalisations of the connection constructed here by looking at variations of the moduli problem of vector bundles on curves. A minor variation is by looking at moduli spaces of G-principal bundles, where G is a semisimple group. One could also equip the curve with marked points and look for parabolic structures of the bundle at these points. All of these can be understood as special cases of the moduli problem of $\mathcal {G}$ -torsors, where $\mathcal {G}$ is a parahoric Bruhat–Tits group scheme over the curve (see, e.g., [44, 28, 17]). We hope to come back to the Hitchin connection in this generality in the near future and expect that the construction developed in this paper, bypassing the need for an explicit description of a Kähler form, will facilitate this.

1.4

The rest of the paper is organised as follows. In Section 2 a summary of Hitchin’s work is given, explaining the context of variation of Kähler polarisation in geometric quantisation. There are essentially two parts to this: a general framework that gives conditions under which a projective connection exists (Theorem 2.1.1) and a discussion of why these conditions are satisfied in the case of moduli spaces of flat unitary connections on surfaces. Though none of what follows later logically depends on this, we nevertheless wanted to include a brief overview of Hitchin’s original construction to highlight the extent to which our exposition parallels his.

The remainder of the paper is then concerned with our algebro-geometric construction of the Hitchin connection. In Section 3, after a quick review of Atiyah sequences and Atiyah classes, the notion of heat operators, their relations to connections and the main framework of van Geemen and de Jong is given (Theorem 3.4.1). We present the latter as a counterpart to Theorem 2.1.1, and for completeness, we have included a proof of it and of Hitchin’s flatness criterion (Theorem 3.5.1) to highlight that these results hold in arbitrary characteristic, as the original discussion in [57] was strictly speaking just in a complex context.

Section 4 then goes on to show that the conditions of Theorem 3.4.1 are indeed satisfied, culminating in Theorem 4.8.1. The primary tool to this end is Proposition 4.7.1, and most of the rest of the section is essentially a (necessarily lengthy) mise en place to obtain this result. As stated above, the key element is Theorem 4.4.1, which realizes the Atiyah class of the determinant-of-cohomology line bundle as a particular extension, given as the first derived functor of the push down of the dual of the Atiyah sequence of the universal bundle on the moduli space of bundles. This provides an analogue to the theorem of Quillen that realizes the Chern class of the line bundle as a particular Kähler form. Just as in Hitchin’s original approach, it is this particular realisation that allows us to verify the cohomological conditions of Theorem 3.4.1. Theorem 4.4.1 is itself obtained from a variation on the theory of the trace complex, of which we give a self-contained account in Appendix A. The proof of Theorem 4.4.1 takes up Section 5. Finally, the other appendices contain proofs of various facts we use in the main body of the article but for which we could not find references in the generality we needed.

1.5

To finish the introduction, we state the necessary restrictions on the characteristic p of the base field $\Bbbk $ and their sources. The first limitation that we encounter is due to the use of the trace and the trace pairing:

We need these to behave similarly as they do in characteristic zero. In particular we want the trace $\operatorname {tr}$ to split equivariantly, i.e., $\mathcal {E} nd(E)=\mathcal {E} nd^0(E)\oplus \mathcal {O}$ , where $\mathcal {E} nd^0(E)$ is the kernel of $\operatorname {tr}$ . This is induced from an $\operatorname {SL}(r)$ -equivariant splitting of the short exact sequence of Lie algebras

which requires $p \nmid r$ . Secondly, we want the trace pairing $\operatorname {Tr}$ , which is nondegenerate for all possible characteristics p and $r=rk(E)$ , to remain nondegenerate when restricted to $\mathcal {E} nd^0(E)\times \mathcal {E} nd^0(E)$ . This is again true if and only if $p \nmid r$ .

The second limitation is due to the use of differential operators (cf. [27, IV, §16.8]) and their symbols: In characteristic $p>0$ one considers the algebra of differential operators associated to the Atiyah algebra $\mathcal {D}^{(1)}_{\mathcal {M}/S}(L)$ and defined as a quotient of its universal enveloping algebra—see [16, 1.1.3]. Up to order $k=p-1$ these, however, coincide with $\mathcal {D}^{(k)}_{\mathcal {M}/S}(L)$ , and we have the symbol map to $\operatorname {\mathrm {Sym}}^k T_{\mathcal {M}/S}$ with its usual properties at our disposal. As the construction of connections via heat operators uses second-order operators and their symbols, we exclude characteristic 2; in the flatness criterion also third-order symbols appear; hence, there we also exclude $p=3$ .

Furthermore, we also use trace complexes; the original reference avoids positive characteristic, but as we use only part of the theory, we check in Appendix A that everything works with the restrictions already in place: In order for the residue $\widetilde {\operatorname {\mathrm {res}}}$ from [16, p. 658] to be well defined, we need to avoid characteristic 2.

The third and last limitation is due to the formula in Theorem 4.8.1, where there is a factor $\frac {1}{r+k}$ . Hence, we also need to assume that $p\nmid (r+k)$ .

1.6 Acknowledgements

The authors would like to thank Jørgen Andersen, Prakash Belkale, Cédric Bonnafé, Najmuddin Fakhruddin, Emilio Franco, Bert van Geemen, Jochen Heinloth, Nigel Hitchin, Gregor Masbaum, Swarnava Mukhopadhyay, Jon Pridham, Brent Pym, Pavel Safronov, Richard Wentworth and Hacen Zelaci for useful conversations and remarks at various stages of this work. This work grew out of another project of the first and third named authors that was joint with Jørgen Andersen, Peter Gothen and Shehryar Sikander—they thank all three of them for related discussions.

2 Heat operators and connections—summary of the work of Hitchin

We outline in this section the original work of Hitchin that establishes the flat projective connection on bundles of nonabelian theta functions. Hitchin’s motivation came from geometric quantisation and Kähler geometry, and he mainly used analytic or Kähler techniques.

2.1 Change of Kähler polarisation

Inspired by earlier work of Welters [58], the Hitchin connection was introduced in [30] in the context of geometric quantisation: Given a compact (real) symplectic manifold $(\mathcal {M},\omega )$ , with prequantum line bundle L, Hitchin studied how the geometric quantisations with respect to different Kähler polarisations were related. In particular, he gave the following general criterion for the existence of a projective connection on the bundle of quantisations:

Theorem 2.1.1 Hitchin, [30, Theorem 1.20]

Given a family of Kähler polarisations on $\mathcal {M}$ such that for each polarisation we have:

1. (a) The map

   
   is an isomorphism (this means that there are no holomorphic vector fields which fix L, i.e., $H^0(\mathcal {M}, \mathcal {D}^{(1)}_{\mathcal {M}}(L))=H^0(\mathcal {M}, \mathcal {O}_{\mathcal {M}})$ ).

2. (b) For each $s\in H^0(\mathcal {M},L)$ and tangent vector $\overset {.}{I}$ to the base of the family there exists a smoothly varying

   $$ \begin{align*} A(\overset{.}{I}, s)\in \mathbb{H}^1(\mathcal{M}, \mathcal{D}^{(1)}_{\mathcal{M}}(L)\overset{.s}{\rightarrow} L) \end{align*} $$
   such that the symbol $-i\sigma _1 (A(\overset {.}{I}, s))$ equals the Kodaira–Spencer class $[\overset {.}{I}]$ in $H^1(\mathcal {M}, T_{\mathcal {M}})$ .

Then this defines a projective connection on the bundle of projective spaces $\mathbb {P}(H^0(\mathcal {M}, L))$ over the base of the family.

Here, $\mathcal {D}^{(1)}_{\mathcal {M}}(L)$ denotes the sheaf of first-order differential operators on L and $\sigma _1$ its symbol map to $T_{\mathcal {M}}$ . The map $.s:\mathcal {D}^{(1)}_{\mathcal {M}}(L)\rightarrow L$ is just given by evaluating the differential operators on the section s, and $\mathbb {H}^1$ stands for the first hypercohomology group of the two-term complex.

Note that the space of infinitesimal deformations of the pair $(\mathcal {M}, L)$ is given by $H^1(\mathcal {M}, \mathcal {D}^{(1)}_{\mathcal {M}}(L))$ , and likewise the space of infinitesimal deformations of the triple $(\mathcal {M},L,s)$ , for $s\in H^0(\mathcal {M},L)$ , is given by $\mathbb {H}^1(\mathcal {M}, \mathcal {D}^{(1)}_{\mathcal {M}}(L)\overset {.s}{\rightarrow } L)$ (cfr. [58, Proposition 1.2]).

2.2 Moduli spaces of flat unitary connections

Moreover, Hitchin then showed that the conditions of Theorem 2.1.1 are satisfied in the case where $(\mathcal {M},\omega )$ is the space of flat, unitary, trace-free connections on the trivial rank r bundle over a closed oriented surface $\mathcal {C}$ of genus $g\geq 2$ (with the exception of the case $r=2, g=2$ ), and $L=\mathcal {L}^k$ is a power of the positive generator $\mathcal {L}$ of its Picard group. This space is not quite a manifold, but its smooth locus is canonically a symplectic manifold, with $\omega $ the Goldman–Karshon symplectic form (which uses a Killing form on the Lie algebra of $\operatorname {SU}(r)$ ).

If $\mathcal {C}$ is equipped with the structure of a Riemann surface (or, equivalently, regarded as a smooth complex projective curve), then $\mathcal {M}$ can be understood as the moduli space of semi-stable rank r vector bundles with trivial determinant, which is a projective variety. The symplectic form $\omega $ is then, moreover, a Kähler form, as discussed by Narasimhan [42] and Atiyah-Bott [1]. By Quillen’s theorem [45], the inverse $\mathcal {L}$ of the determinant-of-cohomology line bundle provides a prequantum line bundle.

In particular, we can understand the $A(\overset {.}{I}, s)$ as follows in this situation: We have the short exact sequence of complexes

(1)

This gives a connecting homomorphism

(2)

On the other hand, the quadratic part of the Hitchin system (which also uses the Killing form) gives, for every holomorphic vector bundle E on $\mathcal {C}$ with trivial determinant, a map

where $K_{\mathcal {C}}$ is the canonical bundle of $\mathcal {C}$ . Dualizing this, and using Serre duality on $\mathcal {C}$ gives, for each E, a map

where $\mathcal {E}nd^0(E)$ is the sheaf of trace-free endomorphisms of E. Since for each stable E the space $H^1(\mathcal {C}, \mathcal {E}nd^0(E))$ is the tangent space to the moduli space (in casu $\mathcal {M}$ ), we can write this as a map

(3)

Composing this with (2) gives a linear map

which depends smoothly on s, and which Hitchin shows (after a rescaling by $\frac {1}{r+k}$ ) to satisfy the condition in (b) of Theorem 2.1.1.

Remark 2.2.1. Some key steps in Hitchin’s approach were fundamentally differential geometric or Kähler in nature. In particular, the explicit description of the Narasimhan–Atiyah–Bott Kähler form, and its similarity to the symmetric two-tensors given by the symbol was crucially used.

3 Hitchin-Type Connections in Algebraic Geometry

An algebro-geometric framework for connections determined by a heat equation (like the Hitchin connection) was developed by van Geemen and de Jong in [57]. Besides being set in algebraic geometry as opposed to Kähler geometry, this description is also more local, in contrast with the infinitesimal framework of Theorem 2.1.1 of Hitchin (the latter is not a substantial difference, however, cfr. [57, §2.3.4]). We summarise the main parts and some related prerequisites below.

From now on, everything will be defined over an algebraically closed field $\Bbbk $ of characteristic different from $2$ . We have to exclude characteristic $2$ for a variety of reasons but, in particular, will also split the projection $T_{\mathcal {M}}^{\otimes 2}\rightarrow \operatorname {\mathrm {Sym}}^2 T_{\mathcal {M}}$ throughout. In this general section, $\mathcal {M} \rightarrow S$ will be a smooth morphism of smooth schemes.

3.1 Atiyah algebroids, (projective) connections and Atiyah classes

Our approach to connections essentially follows Atiyah’s seminal exposition [5], but in this context we will phrase everything in terms of vector bundles rather than work with principal bundles.

Atiyah algebroids

Let $\mathcal {D}^{(n)}_{\mathcal {M}}(E)$ be the sheaf of differential operators of order at most n on a vector bundle E over $\mathcal {M}$ . The associated symbol map will be denoted

$$ \begin{align*} \sigma_n:\mathcal{D}^{(n)}_{\mathcal{M}}(E)\rightarrow \operatorname{\mathrm{Sym}}^n T_{\mathcal{M}}\otimes \mathcal{E}nd(E). \end{align*} $$

Definition 3.1.1. The Atiyah sequence associated to a vector bundle $E\rightarrow \mathcal {M}$ is the top row of the following diagram:

The middle term $\mathcal {A}(E)$ is called the Atiyah algebroid associated to E (or, strictly speaking, to the frame bundle associated to E, which is a $\operatorname {GL}$ -principal bundle).

Definition 3.1.2. We will denote by $\mathcal {A}_{\mathcal {M}/S}(E)$ , the relative Atiyah algebroid associated to a vector bundle $E\rightarrow \mathcal {M}$ , where $\mathcal {M}$ comes with a morphism $\pi : \mathcal {M}\rightarrow S$ onto a base scheme S. The associated relative Atiyah sequence is the top row of the following pull-back diagram:

(4)

where $T_{\mathcal {M}/S}$ is the subsheaf of vector fields tangent along the fibres, i.e.,

$$ \begin{align*}T_{\mathcal{M}/S} = \operatorname{\mathrm{Ker}}(T_{\mathcal{M}} \to \pi^* T_S).\end{align*} $$

Finally, we need to define the trace-free Atiyah algebroid for vector bundles with trivial determinant. Pushing out the standard Atiyah sequence by the trace map $\mathcal {E}nd(E)\rightarrow \mathcal {O}$ gives a morphism of the Atiyah sequene of E to that of $\det (E)$ . If the latter is trivial, its Atiyah sequence splits canonically, giving rise a morphism $\operatorname {\mathrm {tr}}:\mathcal {A}(E)\rightarrow \mathcal {O}$ . We define the trace-free Atiyah algebroid $\mathcal {A}^0(E)$ to be the kernel of this map. This all fits together in a commutative diagram (with exact horizontal rows and left vertical row):

The algebroid $\mathcal {A}^0(E)$ can be understood, in the language of principal bundles, as arising from the $\operatorname {SL}(r)$ -principal frame bundle of E. Analogously, there is also a relative version $\mathcal {A}^0_{\mathcal {M}/S}(E)$ .

Assuming $p \nmid r$ , we have a direct sum decomposition $\mathcal {E} nd(E) = \mathcal {E} nd^0(E) \oplus \mathcal {O}_{\mathcal {M}}$ , and we denote by $q: \mathcal {E} nd(E) \to \mathcal {E} nd^0(E)$ the projection onto the first direct summand. In this case, the trace-free Atiyah algebroid is also canonically isomorphic to the projective Atiyah algebroid, i.e., the push-out of the standard Atiyah sequence by the map q as follows:

We will make this identification throughout.

Atiyah classes

We will also need a relative version of the Atiyah class for a line bundle L. There are a number of ways this can be defined; perhaps the easiest is by taking the top sequence of equation (4), tensoring it with $\Omega ^1_{\mathcal {M}/S}$ , and applying $\pi _*$ to obtain a long exact sequence (of course for line bundles we have canonically $\mathcal {E}nd(L)\cong \mathcal {O}$ ).

Definition 3.1.3. The image of the identity $\pi _{\ast }\operatorname {Id}\in \pi _*\big ( \Omega ^1_{\mathcal {M}/S}\otimes T_{\mathcal {M}/S}\big )$ under the connecting homomorphism yields a global section of $R^1\pi _* \big (\Omega ^1_{\mathcal {M}/S} \otimes \mathcal {E}nd(E)\big )$ , which we shall refer to as the relative Atiyah class, and denote by $[L]$ .

Note that the connecting homomorphism in the long exact sequence obtained by applying $\pi _*$ to the top sequence of equation (4) is given by cupping with $[L]$ and contracting. In the absolute case, the Atiyah class is the obstruction to the existence of a connection on L; a similar interpretation holds in the relative case, though we will not use this. If $\mathcal {M}$ is complex Kähler, $[L]$ is just the relative Chern class.

The following lemma probably dates back to [5]; see, e.g., [36, p. 431].

Lemma 3.1.4. Let X be a smooth algebraic variety, L a line bundle, k a positive integer, then we have an isomorphism of short exact sequences

Projective connections

Definition 3.1.5. Given a vector bundle E on a variety $\mathcal {M}$ , a (Koszul) connection $\nabla $ on E is a $\mathcal {O}_{\mathcal {M}}$ -linear splitting of the Atiyah algebroid:

The connection is said to be flat (or integrable) if $\nabla $ preserves the Lie brackets (where the Lie bracket on $\mathcal {A}(E)$ is just the commutator of differential operators).

The Hitchin connection is a projective connection. There are a number of ways one can encode what a projective connection is: One could think in terms of $\operatorname {PGL}$ principal bundles, or work with the projectivisation $\mathbb {P}(E)$ of E, or work with twisted $\mathcal {D}$ -modules (cfr. [12], [36, §1]). In our context, the most useful one is the following.

Definition 3.1.6. Given a vector bundle E on $\mathcal {M}$ as before, a projective connection is a splitting

It is again flat if $\nabla $ preserves the Lie brackets.

3.2 Heat operators

Consider a smooth surjective morphism of smooth schemes $\pi :\mathcal {M}\rightarrow S$ , and a line bundle $L\rightarrow \mathcal {M}$ such that $\pi _{\ast } L$ is locally free, hence a vector bundle. The connection we construct will live on the projectivisation $\mathbb {P}\pi _{\ast } L$ , but everything below will be expressed in terms of vector bundles, not projective bundles.

We will denote by $\mathcal {D}^{(n)}_{\mathcal {M}/S}(L)$ the subsheaf of $\mathcal {D}^{(n)}_{\mathcal {M}}(L)$ consisting of differential operators of order at most n that are $\pi ^{-1}(\mathcal {O}_S)$ linear. The symbol maps

take values in $\operatorname {\mathrm {Sym}}^n T_{\mathcal {M}/S}$ .

We are now interested in the sheaf

$$ \begin{align*} \mathcal{W}_{\mathcal{M}/S}(L)=\mathcal{D}^{(1)}_{\mathcal{M}}(L)+\mathcal{D}^{(2)}_{\mathcal{M}/S}(L)\subset \mathcal{D}^{(2)}_{\mathcal{M}}(L). \end{align*} $$

Besides the second-order symbol map

on this sheaf of differential operators, there is a subprincipal symbol

(5)

where s is a local section of L and f a local section of $\mathcal {O}_S$ ; both well-definedness and the Leibniz rule follow from the property of the second-order symbol

$$\begin{align*}D(fg s) = \langle \sigma_2(D) , df \otimes dg \rangle s + f D(gs)+g D(fs)-fg D(s). \end{align*}$$

Thus, we have a short exact sequence

(6)

We can now define:

Definition 3.2.1 [57, 2.3.2]

A heat operator D on L is a $\mathcal {O}_S$ -linear map of coherent sheaves

such that $\sigma _S \circ \widetilde {D}=\text {Id}$ , where $\widetilde {D}$ is the equivalent (by adjunction) $\mathcal {O}_{\mathcal {M}}$ -linear map

$$ \begin{align*}\widetilde{D} : \pi^* T_S \to \mathcal{W}_{\mathcal{M}/S}(L).\end{align*} $$

Similarly, a projective heat operator is a map

Given such a heat operator, we refer to

as the symbol of the heat operator. Also, a projective heat operator has a well-defined symbol.

3.3 Heat operators and connections

Any heat operator gives rise to a connection on the locally free sheaf $\pi _* L$ , as follows (cfr. [57, §2.3.3]). Given an open subvariety $U\subset S$ , and $\theta \in T(U)$ , we want a first-order differential operator

If $s\in \pi _* L(U)$ , we denote by s and $\pi ^{-1}(\theta )$ the corresponding sections of $L(\pi ^{-1}(U))$ and $\pi ^{-1}(T_S)(\pi ^{-1}(U))$ , respectively. We can now put

(7) $$ \begin{align} \nabla_{\theta}s=D(\pi^{-1}(\theta))(s) \end{align} $$

since the latter indeed corresponds to a section of $\pi _* L(U)$ . Moreover, the Leibniz rule is satisfied since the subprincipal symbol of $D(\pi ^{-1}\theta )$ is $\pi ^{-1}\theta $ so that for any $f\in \mathcal {O}_S(U)$ we have

$$\begin{align*}\nabla_{\theta}(fs)=D(\pi^{-1}(\theta))(\pi^{\ast}(f) s) = \pi^{\ast}(\theta(f)) s+ \pi^{\ast}(f) D(\pi^{-1}(\theta))(s)= \theta(f) s+ f\nabla_{\theta}s, \end{align*}$$

so $\nabla _{\theta }$ is indeed a first-order differential operator with symbol $\theta $ , and hence, $\nabla $ is indeed a Koszul connection.

The connection $\nabla $ will be flat if D preserves the Lie brackets. If we have a projective heat operator, we still get a projective connection, with the same comment for flatness.

3.4 A heat operator for a candidate symbol

As an algebro-geometric counter-part to Hitchin’s Theorem 2.1.1, van Geemen and de Jong investigated under what conditions a candidate symbol map

actually arises as a symbol of a heat operator, i.e., whether it was possible to find a (projective) heat operator D such that $\rho = \pi _*(\sigma _2) \circ D$ . Before we can state their result, we need to recall two maps. The canonical short exact sequence

gives rise to the Kodaira–Spencer map

(8)

Similarly, the short exact sequence

(9)

gives rise to the connecting homomorphism

(10)

We can now state:

Theorem 3.4.1 van Geemen–de Jong, [57, §2.3.7]

With L and $\pi :\mathcal {M}\rightarrow S$ as before, we have that if, for a given $\rho : T_S \rightarrow \pi _{\ast } \operatorname {\mathrm {Sym}}^2 T_{\mathcal {M}/S}$ ,

1. (a) $\kappa _{\mathcal {M}/S}+\mu _{L} \circ \rho =0,$

2. (b) cupping with the relative Atiyah class

   
   is an isomorphism and

3. (c) $\pi _*\mathcal {O}_{\mathcal {M}}=\mathcal {O}_S$ ,

then there exists a unique projective heat operator D whose symbol is $\rho $ .

Note that even though the context of this theorem is entirely algebro-geometric and makes no reference to a symplectic form, the conditions are closely matched with those in Hitchin’s Theorem 2.1.1: The requirement of cupping with the Chern class being an isomorphism is identical in both cases, whereas from a quadratic symbol $\rho $ satisfying condition (a) we recover an element of the hypercohomology group in 2.1.1.(b) via the long-exact sequence of hypercohomology obtained from equation (1). Finally, (c) is an appropriate weakening of the premise that $\mathcal {M}$ is compact (and connected) in Theorem 2.1.1.

Proof. Consider the long exact sequence associated to the short exact sequence (6),

As $\cup [L]$ is the connecting homomorphism in the long exact sequence associated with the first-order symbol map on $\mathcal {D}^{(1)}_{\mathcal {M}/S}(L)$ , condition (b) guarantees that $\mathcal {O}_S = \pi _{\ast } \mathcal {O}_{\mathcal {M}} = \pi _{\ast } \mathcal {D}^{(1)}_{\mathcal {M}/S}(L)$ , i.e., all global first-order operators on L along the fibres of $\pi $ are of order zero. Using condition (c), we obtain a commutative diagram with exact rows and columns

and therefore an isomorphism $ \left ( \pi _{\ast } \mathcal {W}_{\mathcal {M}/S}(L) \right ) / \mathcal {O}_S \to \operatorname {\mathrm {Ker}} \delta $ . It remains to show that our hypotheses imply that the image of the morphism

is contained in the kernel of the connecting homomorphism $\delta $ . In order to do this, let us decompose $\delta =\delta _1 + \delta _2$ into its two components:

It is then straightforward to check that

$$ \begin{align*}R^1\pi_*(\sigma_1)\circ \delta_1= \kappa_{\mathcal{M}/S}\ \ \ \ \textrm{and}\ \ \ \ R^1\pi_*(\sigma_1)\circ \delta_2= \mu_{L}.\end{align*} $$

Finally, we observe that $\sigma _1$ induces an injective map

as the previous map in the long exact sequence

is surjective by condition (a). Thus, $(\theta , \rho (\theta ))\in \operatorname {\mathrm {Ker}} \delta $ if and only if $(\kappa _{\mathcal {M}/S} + \mu _{L} \circ \rho )(\theta )=0$ , for any local vector field $\theta $ on S.

3.5 A flatness criterion

To complete our outline of the general part of the theory, we discuss a general flatness condition for connections constructed via Theorem 3.4.1. It is a verbatim translation of Hitchin’s original reasoning [30, Thm. 4.9] to the algebro-geometric setting, its central ingredient being the requirement that the symbols should Poisson-commute when viewed as homogeneous functions on the relative cotangent bundle.

Theorem 3.5.1. Under the conditions of Theorem 3.4.1 and over a base field of characteristic different from 3, the projective connection constructed from a symbol $\rho $ is projectively flat if

1. (a) for all local sections $\theta ,\theta '$ of $T_S$ ,

   $$\begin{align*}\{ \rho(\theta), \rho(\theta') \}_{T^{\ast}_{\mathcal{M}/S}} = 0 , \end{align*}$$

2. (b) the morphism $\mu _L$ is injective and

3. (c) there are no vertical vector fields, $\pi _{\ast } T_{\mathcal {M}/S}=0$ .

Remark 3.5.2. In the statement and the proof of this theorem, we use the fact that the natural morphism

$$\begin{align*}\pi_{\ast} \operatorname{\mathrm{Sym}}^k T_{\mathcal{M}/S} \to \pi_{\ast}\mathcal{O}_{T^{\ast}_{\mathcal{M}/S}} \end{align*}$$

is an isomorphism of Poisson algebras onto the weight k part under the natural $\mathbb {G}_m$ -action for $k \leq p-1$ ; here, the Poisson structure on the left is the one inherited from the commutator bracket on operators of order at most k, and the one on the right is the natural one on the cotangent bundle.

Proof. As the connection is defined by projective heat operators (7), its flatness is equivalent to the vanishing of the operator

(11) $$ \begin{align} [D(\theta),D(\theta')]-D([\theta,\theta']) \in \pi_{e\ast} \left( \mathcal{D}^{(3)}_{\mathcal{M}/S}(\mathcal{L}^k)+\mathcal{D}^{(2)}_{\mathcal{M}}(\mathcal{L}^k) \right) \big/ \mathcal{O}_S. \end{align} $$

Now, it follows from the preceding remark and condition (a) that

$$\begin{align*}\sigma_3([D(\theta),D(\theta')]) = \left\{ \sigma_2(D(\theta)),\sigma_2(D(\theta')) \right\}_{T^{\ast}_{\mathcal{M}/S}} = \left\{ \rho(\theta),\rho(\theta') \right\}_{T^{\ast}_{\mathcal{M}/S}} = 0. \end{align*}$$

Therefore, the operator (11) is actually at most second order, and we furthermore claim that it really acts only along the fibres of $\mathcal {M} \rightarrow S$ ,

$$\begin{align*}[D(\theta),D(\theta')]-D([\theta,\theta']) \in \pi_{e\ast} \left( \mathcal{D}^{(2)}_{\mathcal{M}/S}(\mathcal{L}^k) \right) \big/ \mathcal{O}_S. \end{align*}$$

This happens for the same reason the curvature $[\nabla _X,\nabla _Y]-\nabla _{[X,Y]}$ of a connection is of degree zero as a differential operator: One checks (using the subprincipal symbol (5)) that equation (11) is $\pi ^{-1}\mathcal {O}_S$ -linear.

Now, we look at the short exact sequence (9), and apply $\pi _{\ast }$ . As $\mu _L$ is injective by condition (b) and there are no vertical vector fields by (c), we get

$$ \begin{align*}\pi_{\ast}\mathcal{D}^{(2)}_{\mathcal{M}/S}(L)\Big/\mathcal{O}_S \cong \pi_{\ast} T_{\mathcal{M}/S} = 0,\end{align*} $$

thus concluding the proof.

3.6 The map $\mu _{L}$

Finally, we need to get a better understanding of the map $\mu _{L}$ from equation (10), for which we could simply refer to [8, Cor. 2.4.6]. As the proof is not too complicated and uses only a fraction of the machinery of that paper, we thought it worthwile to include it here. We thank an anonymous referee for pointing out considerable simplifications to our previous proof.

Proposition 3.6.1. In the context outlined above (with $\pi :\mathcal {M}\rightarrow S$ is a smooth morphism of smooth schemes and L a line bundle on $\mathcal {M}$ ), we can write the connecting homomorphism (10) as

$$ \begin{align*} \mu_{L}= \cup [L] + \cup\left(-\frac{1}{2} [K_{\mathcal{M}/S}]\right), \end{align*} $$

where $K_{\mathcal {M}/S}$ is the relative canonical bundle of $\pi :\mathcal {M}\rightarrow S$ .

Note that ‘half’ of this statement ( $\mu _{L}=\cup [L]+\mu _{\mathcal {O}_{\mathcal {M}}}$ ) appears in [58, Lemma 1.16], except that Welters uses the extension class of the sheaf of principal parts $\mathcal {P}^{(1)}(L)$ of order ${\leq}1$ instead of $\mathcal {D}^{(1)}_{\mathcal {M}/S}(L)$ to define $[L]$ and hence has a minus sign on the right-hand side. In a Kähler context, with L a polarizing line bundle, the statement of Proposition 3.6.1 is implied in [30, p. 364]. In the general complex analytic setting, a Dolbeault-theoretic approach is descibed in [14, Appendix A.2] ¹ .

Proof. The proof follows from the identification of the opposite of the algebra of differential operators on L with that of $L^{-1}\otimes K_{\mathcal {M}/S}$ via the adjoint differential operator $D^{\circ }$ , as discussed, for example, in [16, 1.1.5.(iv)]. Due to the identity $\mu _{L}=\cup [L]+\mu _{\mathcal {O}_{\mathcal {M}}}$ observed already by Welters (in arbitrary characteristic), it suffices to show that

(12) $$ \begin{align} \mu_{L} = -\mu_{L^{-1}\otimes K}. \end{align} $$

For this, consider the adjoint map between sheaves of differential operators $\mathcal {A}_{\mathcal {M}/S}(E) \ni D \mapsto D^{\circ } \in \mathcal {A}_{\mathcal {M}/S}(E^{\ast } \otimes K_{\mathcal {M}/S})$ defined by the identity

$$\begin{align*}\langle e, D^{\circ} e^{\circ} \rangle = \langle De, e^{\circ} \rangle - \mathcal{L}_{\sigma_1 D}\langle e, e^{\circ} \rangle , \end{align*}$$

where e and $e^{\circ }$ are arbitrary local sections of E and $E^{\circ } := E^{\ast } \otimes K_{\mathcal {M}/S}$ , respectively, and $\mathcal {L}$ is the Lie derivative on the relative canonical bundle. It is straightforward to verify that $D^{\circ }$ has symbol $\sigma _1(D^{\circ }) = - \sigma _1(D)$ and that for any regular local function $\phi $

$$\begin{align*}(\phi D)^{\circ} = \phi D^{\circ} - \langle \sigma_1 D, d_{\mathcal{M}/S}\phi \rangle \end{align*}$$

so that $D\mapsto D^{\circ }$ is in particular $\pi ^{-1}\mathcal {O}_S$ -linear. This zeroth-order deviation from $\mathcal {O}_{\mathcal {M}}$ -linearity may appear inconvenient at first sight, but it actually permits to extend the adjoint to second-order operators, as

$$\begin{align*}(\phi D_2)^{\circ} \circ D_1^{\circ} = D_2^{\circ} \circ (\phi D_1)^{\circ} + (\langle \sigma_1 D_1,d\phi \rangle D_2)^{\circ}. \end{align*}$$

In this way, we obtain a $\pi ^{-1}\mathcal {O}_S$ -linear isomorphism of short exact sequences

whose push-out along $\sigma _1$ gives

which proves the necessary identity (12).

Remark 3.6.2. Note that the preceding result remains true in characteristic $p>0$ with $p\neq 2$ since we only use the isomorphism induced by $D \mapsto D^{\circ }$ between differential operators of order $\leq 2$ .

4 An algebro-geometric approach to the Hitchin connection for nonabelian theta functions

In this section, we construct the Hitchin connection in algebraic geometry. We want to invoke Theorem 3.4.1, using the symbol $\rho $ from equation (3) on page 8. In order to verify that this theorem applies, we need to begin by examining the various ingredients of condition (a).

Note that, compared to the situation of families of abelian varieties (cfr. [58], [57, §2.3.8]), we need a much more detailed knowledge of our candidate symbol in order to establish flatness of the connection later on (which is done via other means for abelian varieties).

4.1 Basic facts about the moduli space of bundles

At this point, we can turn our attention to the particular context we are interested in: the moduli theory of bundles on curves. In the rest of Section 4, we shall denote by $\pi _s:\mathcal {C}\rightarrow S$ a smooth family of smooth projective curves of genus $g\geq 2$ . This gives rise for any integer $r\geq 2$ to a (coarse) relative moduli space of stable bundles of rank r with trivial determinant over the same base, which we shall denote by $\pi _e:\mathcal {M}\rightarrow S$ . If $g=2$ we will assume that $r\geq 3$ . We shall denote the fibred product by the diagram

and will simply put

$$ \begin{align*} \pi_c=\pi_e\circ \pi_n=\pi_s\circ \pi_w. \end{align*} $$

Unfortunately, $\mathcal {M}$ is only a coarse moduli space, and a universal bundle over $\mathcal {C}\times _S \mathcal {M}$ does not exist (one could argue that it exists over the stack of stable bundles $\mathfrak {M}\rightarrow S$ , but does not descend to $\mathcal {M}$ ). Nevertheless, one can speak both of the Atiyah algebroid and Atiyah sequence of the virtual bundle (since these do descend to the coarse moduli space). There exists a unique line bundle $\mathcal {L}$ over $\mathcal {M}$ , called the theta line bundle, which is mapped to the relatively ample generator of the relative Picard variety $\operatorname {\mathrm {Pic}}(\mathcal {M}/S)$ (see [19, 32]). In order to avoid making our notations heavier than needed, we shall henceforth pretend a universal bundle $\mathcal {E}\rightarrow \mathcal {C}\times _S\mathcal {M}$ exist. Note that this universal bundle is only unique up to tensor product with a line bundle coming from $\mathcal {M}$ .However the trace-free endomorphism bundle $\mathcal {E} nd^0(\mathcal {E})$ is unique. Similarly, the determinant-of-cohomology line bundle on $\mathcal {M}$ associated to a universal bundle $\mathcal {E}$ , defined as in [34]

$$\begin{align*}\lambda(\mathcal{E}) := \det R^{\bullet} \pi_{n \ast} (\mathcal{E}) , \end{align*}$$

will depend on the choice of the universal bundle $\mathcal {E}$ . We will use two well-known properties when considering vector bundles with trivial determinant.

• • For any universal bundle $\mathcal {E}$ and any line bundle $\zeta $ on $\mathcal {C} \to S$ of degree $g-1$ , we have the equality [19, 32]

  (13) $$ \begin{align} \mathcal{L}^{-1}=\lambda (\mathcal{E} \otimes \pi_w^*\zeta). \end{align} $$

• • For any universal bundle $\mathcal {E}$ , we have the equalities [38]

  (14) $$ \begin{align} \mathcal{L}^{-2r} = K_{\mathcal{M}/S} = \lambda(\mathcal{E} nd^0(\mathcal{E})). \end{align} $$

At various places, we shall use the trace pairing

to identify $\mathcal {E}nd^0(\mathcal {E})$ with its dual $\mathcal {E}nd^0(\mathcal {E})^*$ .

We will need a few other standard facts about the moduli space $\mathcal {M}$ as well:

Proposition 4.1.1. We have

1. (a) $\pi _{n*}\mathcal {E}nd^0(\mathcal {E})=\{0\}$ ,

2. (b) $T_{\mathcal {M}/S}=R^1\pi _{n*}\mathcal {E}nd^0(\mathcal {E})$ ,

3. (c) $\pi _{e*}T_{\mathcal {M}/S}=\{0\}$ ,

4. (d) $R^1\pi _{e*}\mathcal {O}_{\mathcal {M}}=\{0\}$ .

The first two of these follow from basic deformation theory. For the last two, which are also well-known, we include a proof (due to Hitchin) using the Hitchin system in Appendix C.

4.2 The Kodaira–Spencer Map

Our aim in this section is to give a description of the map

(relating deformations of the curve to deformations of the moduli space) which makes the diagram of sheaves on S

(15)

commute, where $\kappa _{\mathcal {C}/S}$ and $\kappa _{\mathcal {M}/S}$ are the Kodaira–Spencer maps, as in equation (8). This is a line of reasoning that essentially goes back to Narasimhan and Ramanan [43].

On $\mathcal {C}\times _S\mathcal {M}$ we have the trace-free relative Atiyah sequence

(16)

As we have that $\pi _{n*}\left (T_{C\times _S \mathcal {M} \big / \mathcal {M}}\right )=0$ and $R^2\pi _{n*}\mathcal {E}nd^0(\mathcal {E})=0$ , applying $R^1\pi _{n*}$ gives the short exact sequence on $\mathcal {M}$

(17)

In order to describe the Kodaira–Spencer map $\kappa _{\mathcal {M}/S}$ , we need to start from the short exact sequence

which is given (see, e.g., [48, §3.3.3] for the case of a line bundle–vector bundles are a straightforward generalisation of the description there and are discussed in [39, §2.3]) by the pull-back of equation (17) along the map

If we apply $\pi _{e*}$ to this, we obtain finally:

Lemma 4.2.1. The Kodaira–Spencer map $\kappa _{\mathcal {M}/S}$ is given by the composition of $\kappa _{\mathcal {C}/S}$ with $\Phi $ , the connecting homomorphism of equation (17):

4.3 The Hitchin Symbol

We have already briefly encountered the Hitchin symbol in equation (3); we shall clarify the precise definition here in the appropriate relative setting. We start from the quadratic part of the Hitchin system, relative over S and its associated symmetric bilinear form (temporarily denoted B)

Recall that the bilinear form B is, in the explicit description of the relative cotangent bundle via Higgs fields $T^{\ast }_{\mathcal {M} /S} = {\pi _n}_{\ast } ( \mathcal {E}nd^0(\mathcal {E})\otimes K_{\mathcal {C} \times _S \mathcal {M} \big / \mathcal {M}} )$ , given by the trace

$$\begin{align*}B(\phi,\psi) = \operatorname{\mathrm{tr}} (\phi \circ \psi). \end{align*}$$

In particular, it factors further through the symmetric square $\operatorname {\mathrm {Sym}}^2 T_{\mathcal {M}/S}^{\ast }$ . Notice as well that, since we assume the characteristic of the base field to be different from 2, the symmetric square is canonically identified with the symmetric 2-tensors, and in particular there is also a canonical identification

$$\begin{align*}\left( \operatorname{\mathrm{Sym}}^2 T_{\mathcal{M}/S}^{\ast} \right)^{\ast} \cong \operatorname{\mathrm{Sym}}^2 T_{\mathcal{M}/S}. \end{align*}$$

Taking the dual $B^{\ast }$ of B, using Serre duality relative to $\pi _n$ on the domain (where in particular $K_{\mathcal {C} \times _S \mathcal {M} / \mathcal {M}} = \pi _w^{\ast } K_{\mathcal {C} / S}$ ) and pushing down via ${\pi _e}_{\ast }$ we obtain a map ${\pi _e}_{\ast } \left ( B^{\ast } \right )$

Combining this with flat base change

$$\begin{align*}R^1 {\pi_n}_{\ast} \pi_w^{\ast} T_{\mathcal{C} / S} \cong \pi_e^{\ast} R^1 {\pi_s}_{\ast} T_{\mathcal{C} / S} , \end{align*}$$

we make the following definition.

Definition 4.3.1. The Hitchin symbol $\rho ^{\operatorname {Hit}}$ is defined as

The morphism $\rho ^{\operatorname {Hit}}$ is in fact an isomorphism. As we do not need this fact directly, we have relegated it to the appendix; see Lemma C.2.2.

For our purpose of comparing the symbol map with the Kodaira–Spencer morphism in the general context of Theorem 3.4.1, we need the following alternative description: Consider first the surjective evaluation map on $\mathcal {C} \times _S \mathcal {M}$ :

(18)

Dualizing equation (18), we get a morphism

so that swapping the first tensor factor and composing with relative Serre duality for $\pi _n$ we obtain a $\mathcal {O}_{\mathcal {C}\times _S \mathcal {M}}$ -linear morphism

(19)

We also use the trace pairing to identify $\operatorname {Tr}: \mathcal {E}nd^0(\mathcal {E})\overset {\cong }{\to } \mathcal {E}nd^0(\mathcal {E})^*$ . Now, we apply ${\pi _e}_{\ast } \circ R^1{\pi _n}_{\ast }$ to equation (19), and by the isomorphism $R^1\pi _{n*}\mathcal {E}nd^0(\mathcal {E})^*\cong R^1\pi _{n*}\mathcal {E}nd^0(\mathcal {E}) \cong T_{\mathcal {M}/S}$ , the projection formula and base change, we obtain a map

(20)

Lemma 4.3.2. The map (20) coincides with the Hitchin symbol 4.3.1.

Proof. The claimed identity follows from commutativity of the diagram

This follows if we in turn dualize, apply Serre duality, for which

$$\begin{align*}\left( R^1 {\pi_n}_{\ast} (\operatorname{ev}^{\ast} ) \right)^{\ast} = {\pi_n}_{\ast} \left( \operatorname{ev} \otimes \text{Id} \right), \end{align*}$$

(and similarly for the other arrow, where additionally $\operatorname {Tr} = \operatorname {Tr}^{\ast }$ ) and observe that the natural pairing on $\mathcal {E}nd^0(\mathcal {E})^{\ast } \otimes \mathcal {E}nd^0(\mathcal {E})$ coincides with $B \circ (\operatorname {Tr}^{-1} \otimes \text {Id})$ by the definition of B and $\operatorname {Tr}$ .

4.4 The theta line bundle and its Atiyah algebroid

Next, we need some observations about the Atiyah algebroid of the theta line bundle $\mathcal {L}$ (see Section 4.1). We recall that $\mathcal {L}$ is mapped to the ample generator of $\operatorname {\mathrm {Pic}}(\mathcal {M}/S)$ and that $\mathcal {L}$ is related to the determinant-of-cohomology line bundle as in equations (13) and (14).

In this setting, the Atiyah sequence for $\mathcal {L}$ relative to S has a remarkably direct description in terms of the Atiyah sequence of the trace-free relative Atiyah algebroid of $\mathcal {E}$ ,

(21)

Note that, since $\mathcal {E} nd^0(\mathcal {E})$ is uniquely defined, also is $\mathcal {A}^0_{\mathcal {C}\times _S\mathcal {M}\big /\mathcal {M}}(\mathcal {E})$ . Indeed, we have

Theorem 4.4.1. The relative Atiyah sequence of the theta line bundle $\mathcal {L}$ is isomorphic to the first direct image $R^1 \pi _{n \ast }$ of the dual of equation (21):

(22)

For a single fixed curve, this result was stated (without proof) in the announcement [23] (see Theorem 9.1), where it is attributed to Beilinson and Schechtman (even though it does not seem to appear in [16]); it can also be derived from results contained in [50]. We give an independent proof in Section 5.

4.5 A comment on extensions of line bundles

Let X be a scheme, V and L, respectively, a vector and a line bundle on X. Let, moreover, F be an extension of L by V

By taking the dual and tensoring with $V\otimes L$ , we get

Consider now the injective natural map

$$ \begin{align*} \psi: L & \to V^*\otimes V\otimes L \\ \ell & \mapsto \operatorname{Id}_V \otimes \ell. \end{align*} $$

Lemma 4.5.1. There exists a canonical injection $\phi :F\hookrightarrow F^*\otimes V \otimes L$ so that the diagram

(23)

commutes.

Proof. We consider the natural $\mathcal {O}_X$ -linear map $\alpha :F \otimes F \to F\otimes L$ defined by

$$\begin{align*}\alpha(f_1\otimes f_2) = f_1 \otimes \pi(f_2) - f_2 \otimes \pi(f_1) \end{align*}$$

for local sections $f_1,f_2$ of F. Then it is easy to check that the image of $\alpha $ is the subbundle $V\otimes L \subset F\otimes L$ . Now, the map $\alpha $ naturally corresponds to an $\mathcal {O}_X$ -linear map $\phi : F \to F^{\ast } \otimes V \otimes L$ , which can be described locally in terms of a basis of local sections $\{e_i\}$ of F and the dual basis $\{e_i^{\ast }\}$ of $F^{\ast }$ as

$$\begin{align*}\phi(f) = \sum_{i=1}^{\operatorname{\mathrm{rk}} F} \left(e_i^{\ast} \otimes f \otimes \pi(e_i) - e_i^{\ast} \otimes e_i \otimes \pi(f) \right). \end{align*}$$

It is now straightforward to check that this $\phi $ makes the above diagram commute.

4.6 Locally freeness of $\pi _{e*}(\mathcal {L})$

We will be assuming that the direct image $\pi _{e*}(\mathcal {L}^k)$ on S is locally free. In characteristic zero, this follows trivially from Kodaira vanishing, but in positive characteristic, it is not known in general (but of course it will always trivally be true for large enough k). For $r=2$ , this is, however, proven in [41].

Note that, in characteristic zero, a coherent sheaf with a flat projective connection will necessarily be locally free, but this need not be true in general.

4.7 The relation between $\rho ^{\operatorname {Hit}}, \Phi $ and $\mathcal {L}$

We can now state the final ingredient we will need to prove the existence of the Hitchin connection:

Proposition 4.7.1. The sheaf morphism $\Phi $ from equation (15) equals minus the composition $(\cup [\mathcal {L}]) \circ \rho ^{\operatorname {Hit}}$ of the Hitchin symbol and the characteristic class $[\mathcal {L}]$ , i.e., the following diagram of sheaves on S commutes:

Proof. We begin with the trace-free Atiyah sequence on $\mathcal {C}\times _S\mathcal {M}$ for $\mathcal {E}$ , relative to $\pi _n$ , as introduced in Section 3.1. To keep the notation light, we shall denote in this proof the Atiyah algebroid $\mathcal {A}^0_{\mathcal {C}\times _S \mathcal {M}\big /\mathcal {M}}(\mathcal {E})$ simply by $\mathcal {A}$ . By using the evaluation maps, as in equation (18), dualizing and tensoring with $\pi ^*_wT_{\mathcal {C}/S}\otimes \mathcal {E}nd^0(\mathcal {E})$ , we obtain the following natural map of exact sequences:

(24)

By relative Serre duality for $\pi _n$ , the lower exact sequence is equal to the following:

(25)

By plugging $V=\mathcal {E}nd^0(\mathcal {E})$ , $L=\pi _w^*T_{\mathcal {C}/S}$ and $F=\mathcal {A}$ in Lemma 4.5.1, we get a map of exact sequences

(26)

Hence, by composing the short exact sequence maps (26) and (24) and using the isomorphism of the target exact sequence with that of equation (25), we get a new map of exact sequences:

(27)

By taking the direct image $R^1\pi _{n*}$ of both sequences, they remain exact and we obtain the commutative diagram

(28)

We now apply $\pi _{e*}$ to both exact sequences in equation (28). The claimed equality is proven once we consider the commutative diagram given by the connecting homomorphisms:

(29)

Since the bottom row of equation (28) is given by tensoring equation (22) by $R^1\pi _{n*}\mathcal {E}nd^0(\mathcal {E})$ , by Theorem 4.4.1 the connecting homomorphism for the bottom row is given by the relative Atiyah class of $\mathcal {L}$ . By Lemma 4.3.2, the left vertical map is given by the Hitchin symbol $\rho ^{\operatorname {Hit}}$ . Since the upper exact sequence of equation (27) is the same as the sequence (16) but with one sign changed (as in equation (23)), by Lemma 4.2.1 the connecting homomorphism for the top row of equation (29) is given by $-\Phi $ .

4.8 Existence and flatness of the connection

We can now summarize the algebro-geometric construction of the Hitchin connection:

Theorem 4.8.1. Let k be a positive integer. Suppose a smooth family $\pi _{e}:\mathcal {C}\rightarrow S$ of projective curves of genus $g\geq 2$ (and $g\geq 3$ if $r=2$ ) is given as before, defined over an algebraically closed field of characteristic different from $2$ , not dividing r and $k+r$ and such that $\pi _{e*}(\mathcal {L}^k)$ is locally free. Then there exists a unique projective connection on the vector bundle $\pi _{e*}(\mathcal {L}^{k})$ of nonabelian theta functions of level k, induced by a heat operator with symbol

$$ \begin{align*} \rho=\frac{1}{r+k}\,\left(\rho^{\operatorname{Hit}}\circ \kappa_{\mathcal{C}/S}\right). \end{align*} $$

Proof. We establish the existence of the projective connection by invoking Theorem 3.4.1 for the line bundle $\mathcal {L}^{k}$ over $\mathcal {M}$ . We recall from equation (14) the equality $K_{\mathcal {M}/S}=\mathcal {L}^{-2r}$ . From Proposition 3.6.1 we therefore have that

$$ \begin{align*} \mu_{\mathcal{L}^k}=\cup(r+k)[\mathcal{L}], \end{align*} $$

and hence, (using Proposition 4.7.1 and equation (15)) we have

$$ \begin{align*} \mu_{\mathcal{L}^k}\circ\rho=\mu_{\mathcal{L}^k}\circ \frac{1}{r+k}\,\left(\rho^{\operatorname{Hit}}\circ \kappa_{\mathcal{C}/S}\right)=\left(\cup[\mathcal{L}]\right)\circ \rho^{\operatorname{Hit}}\circ \kappa_{\mathcal{C}/S}=-\Phi\circ \kappa_{\mathcal{C}/S} = -\kappa_{\mathcal{M}/S}, \end{align*} $$

which establishes condition (a) of Theorem 3.4.1. Condition (b) is trivially satisfied because of Proposition 4.1.1, and condition (c) follows from the algebraic Hartogs’s theorem [56, Lemma 11.3.11], together with the well-known fact that the relative coarse moduli space $\mathcal {M}^{\operatorname {ss}}$ of semi-stable bundles with trivial determinant (which is singular but normal) is proper over S, and if $g>2$ or $r>2$ , the complement of $\mathcal {M}$ will have codimension greater than one in $\mathcal {M}^{\operatorname {ss}}$ .

As for the curvature of the connection, we have:

Theorem 4.8.2. Suppose furthermore that the characteristic of the base field is different from 3. Then the projective connection constructed in Theorem 4.8.1 is flat.

Proof. We apply Theorem 3.5.1: Condition (a) holds since by definition of the Hitchin symbol the corresponding homogeneous functions on $T^{\ast }_{\mathcal {M}/S}$ are the quadratic components of the Hitchin system and, hence, Poisson commute,

$$\begin{align*}\left\{ \rho^{\operatorname{Hit}}(\theta),\rho^{\operatorname{Hit}}(\theta') \right\}_{T^{\ast}_{\mathcal{M}/S}} = 0. \end{align*}$$

Condition (b) is satisfied as $\mu _{\mathcal {L}^k}$ is injective (see Lemma C.2.7 in Appendix C), and (c) holds by Proposition 4.1.1.

5 Proof of Theorem 4.4.1

We shall need the theory of the trace complex, due to Beilinson and Schechtman, or rather a variation thereon due to Bloch and Esnault—see [16] and [9]. In Appendix A, a summary of this theory is given, and we refer to it for definitions of the complexes ${}^{tr\!\!}{\mathcal {A}}^{\bullet }$ , $\mathcal {B}^{\bullet }$ and ${}^{0}{\mathcal {B}}^{\bullet }$ . We will be applying the trace complex in our particular setting here, where $\mathcal {M}$ is as in Section 4.1, $\mathcal {X} = \mathcal {C} \times _S \mathcal {M}$ and $f = \pi _n$ . In this context, we find that the trace complex simplifies significantly, to give Theorem 4.4.1.

Before proving Theorem 4.4.1 we need to prove a few auxiliary results.

Lemma 5.0.1. Following the above notation:

1. (a) the direct image $\pi _{n*}{}^0{\mathcal {B}}^0(\mathcal {E})$ equals 0;

2. (b) the natural map $R^1\pi _{n*}\mathcal {E}nd^0 (\mathcal {E}) \to R^1\pi _{n*}{}^0{\mathcal {B}}^0(\mathcal {E})$ is zero.

Proof. Recall from Section A.2.2 that we have a short exact sequence

By applying the direct image $\pi _{n*}$ , we get

Now, by Proposition 4.1.1 $(a)$ and $(b)$ , $\pi _{n*}\mathcal {E} nd^0(\mathcal {E})=0$ , and the map $T_{\mathcal {M}/S} \to R^1\pi _{n*}\mathcal {E} nd^0(\mathcal {E})$ is an isomorphism. The two claims follow.

Proposition 5.0.2. There exists an isomorphism $\phi : R^1\pi _{n*}{}^0\mathcal {B}^{-1}(\mathcal {E}) \to R^0\pi _{n*}\mathcal {B}^{\bullet }(\mathcal {E} nd^0(\mathcal {E}))$ that makes the following diagram commute.

In particular, $\phi $ induces $2r\cdot \operatorname {Id}_{\mathcal {O}_{\mathcal {M}}}$ on $\mathcal {O}_{\mathcal {M}}$ .

This proposition is already proved by combining [50, Thm. 3.7 and Cor. 3.12]. For the sake of self-containedness, here, we give a complete but slightly different proof of this statement.

Proof. We construct $\phi $ in several steps, notably as the composition of three maps. First of all, let us define a map

For the sake of clarity, we recall the definition of the $0^{th}$ direct image $R^0\pi _{n*}{}^0\mathcal {B}^{\bullet }(\mathcal {E}).$ We choose an acyclic resolution of the complex ${}^0\mathcal {B}^{\bullet }(\mathcal {E})$ as follows

We push this diagram forward through $\pi _n$ and consider the following one:

Remark that the lower horizontal arrow factors as

$$ \begin{align*}R^1\pi_{n*}{}^0\mathcal{B}^{-1}(\mathcal{E}) \to R^1\pi_{n*} \mathcal{E}nd^0(\mathcal{E}) \to R^1\pi_{n*}{}^0\mathcal{B}^{0}(\mathcal{E}).\end{align*} $$

By definition, we have that $R^0\pi _{n*}{}^0\mathcal {B}^{-1}(\mathcal {E}):= \operatorname {\mathrm {Ker}}(B)/ \operatorname {\mathrm {Im}}(A)$ , where

Hence, we can define a map

$$ \begin{align*} \tilde{\phi}: \pi_{n*}\mathcal{C}^1({}^0\mathcal{B}^{-1}(\mathcal{E})) & \to \operatorname{\mathrm{Ker}}(B);\\ \beta & \mapsto (\alpha, \beta), \end{align*} $$

where $\alpha \in \pi _{n*}\mathcal {C}^0({}^0\mathcal {B}^0(\mathcal {E}))$ is uniquely defined by the formula $d_0(\alpha )=\delta ^1(\beta )$ . In fact, we observe that Lemma 5.0.1 implies that $d_0$ is injective and that $\operatorname {\mathrm {Im}}(\delta ^1) \subseteq \operatorname {\mathrm {Im}}(d^0)$ . The map $\tilde {\phi }$ descends to the first of our three maps:

$$ \begin{align*} \phi_1: R^1\pi_{n*}{}^0\mathcal{B}^{-1}(\mathcal{E}) & \to R^0\pi_{n*}{}^0\mathcal{B}^{\bullet}(\mathcal{E});\\ \bar{\beta} & \mapsto \overline{(\alpha,\beta)}, \end{align*} $$

where the overline should be intended as just taking the corresponding classes.

The second map is defined as follows (see Appendix B for the precise definitions of $\widehat {\mathrm {ad}}$ and $\widetilde {\mathrm {ad}}$ ):

$$ \begin{align*} \phi_2: R^0\pi_{n*}{}^0\mathcal{B}^{\bullet}(\mathcal{E}) & \to R^0\pi_{n*}({}^0\mathcal{B}^{-1}(\mathcal{E}nd^0(\mathcal{E})) \to \mathcal{B}^0(\mathcal{E}nd^0(\mathcal{E})));\\ \overline{(\alpha,\beta)} & \mapsto (\widetilde{\mathrm{ad}}(\alpha), \widehat{\mathrm{ad}}(\beta)), \end{align*} $$

where we abuse once more of the notation (and of the reader’s patience) by denoting by $\widehat {\mathrm {ad}}$ and $\widetilde {\mathrm {ad}}$ also the maps on the direct images. Note also that here we consider $\widetilde {\mathrm {ad}}$ as defined on the quotient ${}^0\mathcal {B}^0(\mathcal {E})$ of the subsheaf $\mathcal {B}^0(\mathcal {E})\subset \mathcal {A}(\mathcal {E})$ , and we are allowed to do so since the trivial sheaf is in $\operatorname {\mathrm {Ker}}(\widetilde {\mathrm {ad}})$ . Moreover, we can consider $\mathcal {B}^0(\mathcal {E}nd^0(\mathcal {E}))$ as the target space of $\widetilde {\mathrm {ad}}$ the image of ${}^0\mathcal {B}^0(\mathcal {E})$ via $\widetilde {\mathrm {ad}}$ is contained in $\mathcal {B}^0(\mathcal {E}nd^0(\mathcal {E}))\subset \mathcal {A}(\mathcal {E}nd^0(\mathcal {E}))$ .

The third map is induced on $R^0\pi _{n*}({}^0\mathcal {B}^{-1}(\mathcal {E}nd^0(\mathcal {E})) \to \mathcal {B}^0(\mathcal {E}nd(\mathcal {E})))$ by the natural inclusion ${}^0\mathcal {B}^{-1}(\mathcal {E}nd^0(\mathcal {E})) \hookrightarrow \mathcal {B}^{-1}(\mathcal {E}nd(\mathcal {E}))$ . Hence, this gives a natural map

It is a standard check that these three maps are well defined and pass to the quotient in cohomology.

The situation is now the following: We have two exact sequences and a map $\phi := \phi _3\circ \phi _2 \circ \phi _1$ between extensions:

Now, suppose we have a class $\bar {\beta }$ in $R^1\pi _{n*}{}^0\mathcal {B}^{-1}(\mathcal {E})$ , and let us consider $\beta $ a local section of $\pi _{n*}\mathcal {C}^1({}^0\mathcal {B}^{-1}(\mathcal {E}))$ representing $\bar {\beta }$ . If we denote as above by $\alpha \in \pi _{n*}\mathcal {C}^0({}^0\mathcal {B}^0(\mathcal {E}))$ the uniquely defined local section as in the definition of $\tilde {\phi }$ , then $\phi $ sends $\beta $ on $\overline {(\widetilde {\mathrm {ad}}(\alpha ),\widehat {\mathrm {ad}}(\beta ))}$ .

By Proposition B.0.3 we have a commutative diagram

which implies the claim about the restriction of $\phi $ to $\mathcal {O}_{\mathcal {M}}$ . Thus, $\phi $ also descends to a $\mathcal {O}_{\mathcal {X}}$ -linear map $\phi ^T: T_{\mathcal {M}/S} \to T_{\mathcal {M}/S}.$ Remark in fact that, again by Appendix B and the observations on $\widetilde {\mathrm {ad}}$ made here above, $\phi ^T$ is induced by the adjoint map between the following exact sequences.

Proof of Theorem 4.4.1

The isomorphism of exact sequences claimed in the theorem will follow by composing the following isomorphisms. In the diagram below, they will be composed vertically from the first to the fifth. First, we apply $R^1\pi _{n*}$ to the second identification from Theorem A.2.6. Then we compose with the map from Proposition 5.0.2. The third map is the isomorphism from Theorem A.2.4 applied to $\mathcal {E} nd^0(\mathcal {E})$ (recall that $\lambda (\mathcal {E} nd^0(\mathcal {E})) = \mathcal {L}^{-2r}$ ). The fourth and fifth maps are the canonical isomorphism $\mathcal {A}(\mathcal {L}^{-1}) \cong \mathcal {A}(\mathcal {L}^{-2r})$ obtained by scaling appropriately the extension as in Lemma 3.1.4 with $k=2r$ and $L=\mathcal {L}^{-1}$ . Finally, the last vertical isomorphism $\mathcal {A}(\mathcal {L}^{-1}) \to \mathcal {A}(\mathcal {L})$ is the canonical map between the Atiyah algebra of $\mathcal {L}^{-1}$ and its dual $\mathcal {L}$ (with the opposite symbol map). Hence, we obtain the following commutative diagram:

Note that the first vertical right-hand-side map is $-\operatorname {Tr}$ . This means that the extension class defining the upper short exact sequence is equal to the standard Atiyah sequence of $\mathcal {L}$ as claimed in the theorem.