1 Introduction
1.1
The Hitchin connection was originally introduced in [Reference Hitchin30], 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 [Reference Witten59, Reference Atiyah6]. 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 [Reference Hitchin31, §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 [Reference Welters58], 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, [Reference Schechtman and Varchenko51] 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 [Reference Grothendieck27].
Key elements in our construction are a framework for connections coming from heat operators in algebraic geometry, due to van Geemen and de Jong [Reference van Geemen and de Jong57], as well as a substitute for the Narasimhan–Atiyah–Bott Kähler form [Reference Narasimhan42, Reference Atiyah and Bott1], which according to Quillen [Reference Quillen45] 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 [Reference Hitchin30] 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 [Reference Beĭlinson and Schechtman16], [Reference Bloch and Esnault9] and some additional inputs worked out in [Reference Sun and Tsai50], describing the behaviour of the above objects when replacing a universal bundle by its endomorphism bundle. We observe that the paper [Reference Sun and Tsai50] also describes a construction of the Hitchin connection, but the strategy in [Reference Sun and Tsai50] is different from ours: They construct the Hitchin connection by relying on another argument from [Reference Faltings21], 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 [Reference van Geemen and de Jong57] 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 [Reference van Geemen and de Jong57, §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., [Reference Faltings21, Reference Ramadas46, Reference Ginzburg23, Reference Ran47, Reference Sun and Tsai50, Reference Ben-Zvi and Frenkel18]. 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 [Reference Laszlo, Pauly and Sorger37] or projective flatness of strange duality maps [Reference Belkale11] have been proved with representation-theoretical methods, more precisely by using its equivalence, due to Laszlo [Reference Laszlo35], with the Tsuchiya-Ueno-Yamada (TUY)/Wess-Zumino-Witten (WZW) connection on spaces of conformal blocks [Reference Tsuchiya, Ueno and Yamada55, Reference Tsuchimoto54]. For most of the cited works, the relationship with conformal blocks is undeveloped (they have of course other motivations: e.g., [Reference Sun and Tsai50], which together with [Reference Ginzburg23] 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., [Reference Andersen, Gammelgaard and Lauridsen3, Reference Axelrod, Della Pietra and Witten2, Reference Scheinost and Schottenloher49].
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 [Reference Belkale11], 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 [Reference Katz33], 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 [Reference Brylinski and McLaughlin13, §7]) that the Hitchin connection would have finite monodromy. However, it was shown by Masbaum in [Reference Masbaum40] 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 [Reference Andersen and Ueno7] 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 [Reference Laszlo, Pauly and Sorger37]. 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. [Reference Gilmer22, Reference Gilmer and Masbaum24, Reference Gilmer and Masbaum25, Reference Gilmer and Masbaum26], 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., [Reference Pappas and Rapoport44, Reference Heinloth28, Reference Balaji and Seshadri17]). 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 [Reference van Geemen and de Jong57] 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. [Reference Grothendieck27, 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 [Reference Beĭlinson and Schechtman16, 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 [Reference Beĭlinson and Schechtman16, 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 [Reference Welters58], the Hitchin connection was introduced in [Reference Hitchin30] 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, [Reference Hitchin30, Theorem 1.20]
Given a family of Kähler polarisations on
$\mathcal {M}$
such that for each polarisation we have:
-
(a) The map
is an isomorphism (this means that there are no holomorphic vector fields which fix L, i.e.,
$H^0(\mathcal {M}, \mathcal {D}^{(1)}_{\mathcal {M}}(L))=H^0(\mathcal {M}, \mathcal {O}_{\mathcal {M}})$
).
-
(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 such that the symbol
$$ \begin{align*} A(\overset{.}{I}, s)\in \mathbb{H}^1(\mathcal{M}, \mathcal{D}^{(1)}_{\mathcal{M}}(L)\overset{.s}{\rightarrow} L) \end{align*} $$
$-i\sigma _1 (A(\overset {.}{I}, s))$
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. [Reference Welters58, 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 [Reference Narasimhan42] and Atiyah-Bott [Reference Atiyah and Bott1]. By Quillen’s theorem [Reference Quillen45], 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
This gives a connecting homomorphism
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
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 [Reference van Geemen and de Jong57]. 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. [Reference van Geemen and de Jong57, §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 [Reference Atiyah5], 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:
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 [Reference Atiyah5]; see, e.g., [Reference Looijenga36, 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. [Reference Beĭlinson and Kazhdan12], [Reference Looijenga36, §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
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
We can now define:
Definition 3.2.1 [Reference van Geemen and de Jong57, 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. [Reference van Geemen and de Jong57, §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
$$ \begin{align} \nabla_{\theta}s=D(\pi^{-1}(\theta))(s) \end{align} $$
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
Similarly, the short exact sequence
gives rise to the connecting homomorphism
We can now state:
Theorem 3.4.1 van Geemen–de Jong, [Reference van Geemen and de Jong57, §2.3.7]
With L and
$\pi :\mathcal {M}\rightarrow S$
as before, we have that if, for a given
$\rho : T_S \rightarrow \pi _{\ast } \operatorname {\mathrm {Sym}}^2 T_{\mathcal {M}/S}$
,
-
(a)
$\kappa _{\mathcal {M}/S}+\mu _{L} \circ \rho =0,$
-
(b) cupping with the relative Atiyah class
is an isomorphism and -
(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 [Reference Hitchin30, 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
-
(a) for all local sections
$\theta ,\theta '$
of
$T_S$
,
$$\begin{align*}\{ \rho(\theta), \rho(\theta') \}_{T^{\ast}_{\mathcal{M}/S}} = 0 , \end{align*}$$
-
(b) the morphism
$\mu _L$
is injective and -
(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
$$ \begin{align} [D(\theta),D(\theta')]-D([\theta,\theta']) \in \pi_{e\ast} \left( \mathcal{D}^{(3)}_{\mathcal{M}/S}(\mathcal{L}^k)+\mathcal{D}^{(2)}_{\mathcal{M}}(\mathcal{L}^k) \right) \big/ \mathcal{O}_S. \end{align} $$
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 [Reference Beĭlinson and Bernstein8, 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 [Reference Welters58, 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 [Reference Hitchin30, p. 364]. In the general complex analytic setting, a Dolbeault-theoretic approach is descibed in [Reference Boer14, Appendix A.2]Footnote
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 [Reference Beĭlinson and Schechtman16, 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
$$ \begin{align} \mu_{L} = -\mu_{L^{-1}\otimes K}. \end{align} $$
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. [Reference Welters58], [Reference van Geemen and de Jong57, §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 [Reference Drezet and Narasimhan19, Reference Hoffmann32]). 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 [Reference Knudsen and Mumford34]
$$\begin{align*}\lambda(\mathcal{E}) := \det R^{\bullet} \pi_{n \ast} (\mathcal{E}) , \end{align*}$$
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 [Reference Drezet and Narasimhan19, Reference Hoffmann32] (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 [Reference Laszlo and Sorger38] (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
-
(a)
$\pi _{n*}\mathcal {E}nd^0(\mathcal {E})=\{0\}$
, -
(b)
$T_{\mathcal {M}/S}=R^1\pi _{n*}\mathcal {E}nd^0(\mathcal {E})$
, -
(c)
$\pi _{e*}T_{\mathcal {M}/S}=\{0\}$
, -
(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
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 [Reference Narasimhan and Ramanan43].
On
$\mathcal {C}\times _S\mathcal {M}$
we have the trace-free relative Atiyah sequence
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
$\mathcal {M}$
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., [Reference Sernesi48, §3.3.3] for the case of a line bundle–vector bundles are a straightforward generalisation of the description there and are discussed in [Reference Martinengo39, §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
$\mathcal {C} \times _S \mathcal {M}$
:
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
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
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
$\mathcal {E}$
,
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):
For a single fixed curve, this result was stated (without proof) in the announcement [Reference Ginzburg23] (see Theorem 9.1), where it is attributed to Beilinson and Schechtman (even though it does not seem to appear in [Reference Beĭlinson and Schechtman16]); it can also be derived from results contained in [Reference Sun and Tsai50]. 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
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 [Reference Mehta and Ramadas41].
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:
By relative Serre duality for
$\pi _n$
, the lower exact sequence is equal to the following:
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
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:
By taking the direct image
$R^1\pi _{n*}$
of both sequences, they remain exact and we obtain the commutative diagram
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:
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 [Reference Vakil56, 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 [Reference Beĭlinson and Schechtman16] and [Reference Bloch and Esnault9]. 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:
-
(a) the direct image
$\pi _{n*}{}^0{\mathcal {B}}^0(\mathcal {E})$
equals 0; -
(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 [Reference Sun and Tsai50, 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.
Appendix A The trace complex, following Beilinson–Schechtman and Bloch–Esnault
We give here a presentation of the parts of the theory of trace complexes (due to Beilinson and Schechtman [Reference Beĭlinson and Schechtman16, §2], see also [Reference Esnault and Tsai20]) that we need. We then describe an alternative approach to the trace complexes, suggested by Bloch and Esnault [Reference Bloch and Esnault9, §5.2].
In fact, to suit our purposes, we make two minor variations: First, we make some small changes to ensure that the construction works in positive characteristic (apart from 2), and second, we phrase everything in a relative context. The latter is trivial on a technical level, but we do it as the Bloch–Esnault approach requires an extra condition, which, when we invoke it in the main part of the article, is only satisfied in a relative setting.
Section A.1 below covers the original trace complex and is just expository. In Section A.2, where the alternative of Bloch–Esnault is explained, we also give proofs for various assertions merely stated in [Reference Bloch and Esnault9].
For the purpose of this appendix, we consider a family of smooth projective curves
$f: \mathcal {X} \to \mathcal {M}$
of genus
$g\geq 2$
, relative to a smooth base scheme S,
together with a vector bundle
$\mathcal {E} \to \mathcal {X}$
. We shall write
$\mathcal {E}^{\circ }$
for
$\mathcal {E}^{\ast } \otimes K_{\mathcal {X}/\mathcal {M}}$
.
The trace complex we are interested in describes the Atiyah algebroid
$\mathcal {A}_{\mathcal {M}/S}(\det R^{\bullet } f_{\ast } \mathcal {E})$
(remark that our notation differs from Beilinson and Schechtman’s: our
$\mathcal {M}$
is their S, and our S is just a point in [Reference Beĭlinson and Schechtman16]).
A.1 The Beilinson–Schechtman trace complex
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{\bullet }(\mathcal {E})$
A.1.1 Overview
The relative tangent bundle
$T_{\mathcal {X}/S}$
contains as subsheaves
$T_{\mathcal {X}/\mathcal {M}} \subset T_{f/S} \subset T_{\mathcal {X}/S}$
, where (with
$df: T_{\mathcal {X}/S}\rightarrow f^*T_{\mathcal {M}/S}$
)
$$\begin{align*}T_{f/S} := (df)^{-1} f^{-1}T_{\mathcal{M}/S} , \end{align*}$$
and corresponding Atiyah algebroids
$$ \begin{align*} \mathcal{A}_{\mathcal{X}/\mathcal{M}}(\mathcal{E}) \hookrightarrow \mathcal{A}_{f/S}(\mathcal{E}) \hookrightarrow \mathcal{A}_{\mathcal{X}/S}(\mathcal{E}). \end{align*} $$
The Beilinson–Schechtman trace complex is a three-term complex:
where
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{-2}(\mathcal {E})=\mathcal {O}_{\mathcal {X}}$
,
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{0}(\mathcal {E}) =\mathcal {A}_{f/S}(\mathcal {E})$
and
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{{-1}}(\mathcal {E})$
is an extension (to be defined below in Section A.1.2))
which fits into the following commutative diagram:
The main use of the trace complex
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{\bullet }(\mathcal {E})$
is the following:
Theorem A.1.1 [Reference Beĭlinson and Schechtman16, Thm. 2.3.1]
The relative Atiyah sequence of the determinant-of-cohomology line bundle
$$ \begin{align*} \lambda(\mathcal{E}) = \det R^{\bullet} f_{\ast} \mathcal{E} := \det f_{\ast} \mathcal{E}\otimes \left(\det R^1 f_{\ast} \mathcal{E}\right)^{*} \end{align*} $$
of
$\mathcal {E}$
with respect to f is canonically isomorphic to the short exact sequence
A.1.2 Construction of
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{-1}(\mathcal {E})$
Let
$\Delta \cong \mathcal {X} \subset \mathcal {X} \times _{\mathcal {M}} \mathcal {X}$
denote the diagonal and
$p_1$
and
$p_2$
the two projections of
$\mathcal {X} \times _{\mathcal {M}} \mathcal {X}$
to
$\mathcal {X}$
. For each of the projections
$p_1, p_2$
, we have a residue map
$\operatorname {Res}^1, \operatorname {Res}^2$
along the fibres (cfr. [Reference Tate52, Reference Beĭlinson10, Reference Braunling15]). The following is a key ingredient for us:
Lemma A.1.2 [Reference Beĭlinson and Schechtman16, §2.1.1.1]
There exists a map
$$ \begin{align*} \widetilde{\operatorname{Res}}: K_{\mathcal{X}/\mathcal{M}}\boxtimes K_{\mathcal{X}/\mathcal{M}}(3\Delta) \rightarrow \mathcal{O}_{\mathcal{X}}, \end{align*} $$
which vanishes on
$K_{\mathcal {X}/\mathcal {M}}\boxtimes K_{\mathcal {X}/\mathcal {M}}(\Delta )$
, is symmetric with respect to transposition and such that
$d\widetilde {\operatorname {Res}}=\operatorname {Res}^1-\operatorname {Res}^2$
. The restriction of
$\widetilde {\operatorname {\mathrm {res}}}$
to
$K_{\mathcal {X}/\mathcal {M}}\boxtimes K_{\mathcal {X}/\mathcal {M}}(2\Delta )$
gives a short exact sequence
where the second map is
$\widetilde {\operatorname {\mathrm {res}}}$
and coincides with the restriction to the diagonal
$\Delta $
.
We shall also need a particular description of the sheaf of (relative) first-order differential operators
$\mathcal {D}^{(1)}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
(see [Reference Beĭlinson and Schechtman16, 2.1.1.2] or the introduction of [Reference Esnault and Tsai20], from which we borrow the notation). Here and in what follows, we identify sheaves supported on the diagonal
$\Delta $
with sheaves on
$\mathcal {X}$
. The next lemma is easily deduced from the definition of the “pole at
$\Delta $
” map.
Lemma A.1.3. The symbol short exact sequence for first-order differential operators on
$\mathcal {E}$
relative to f is isomorphic to the exact sequence
where
$\delta $
is the “pole at
$\Delta $
” map defined by
$$ \begin{align*}\delta(\psi)(e) = \operatorname{Res}^2(\langle \psi, p_2^*(e) \rangle),\end{align*} $$
for any local section
$\psi $
of
$\frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ } (2\Delta )}{\mathcal {E} \boxtimes \mathcal {E}^{\circ }}$
and any local section e of
$\mathcal {E}$
. Here,
$\langle -,- \rangle $
is the natural pairing
$\mathcal {E}^{\circ } \times \mathcal {E} \to K_{\mathcal {X}/\mathcal {M}}$
.
We consider now the natural exact sequence
Then the construction that defines the short exact sequence (30) is obtained by taking first the pull-back of equation (33) to
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})\subset \mathcal {D}^{(1)}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
and then the push-out under the trace map
$\mathcal {E} nd(\mathcal {E})\otimes K_{\mathcal {X}/\mathcal {M}} \stackrel {\operatorname {Tr}}{\to } K_{\mathcal {X}/\mathcal {M}}$
,
A.2 The quasi-isomorphic Bloch–Esnault complex
$\mathcal {B}^{\bullet }$
Following [Reference Bloch and Esnault9], we will now construct a subcomplex
$\mathcal {B}^{\bullet }(\mathcal {E}) \subset {}^{\operatorname {tr}\!\!}{\mathcal {A}}^{\bullet }(\mathcal {E})$
that allows for more handy computations. Its construction relies on the existence of a splitting of the short exact sequence
Remark A.2.1. Note that this condition is in particular satisfied whenever
$\mathcal {X}$
is a fibred product
$\mathcal {X} = \mathcal {Y} \times _{S} \mathcal {M}$
and
$f = \pi _2$
the projection since then
$T_{\mathcal {X}/S} \cong \pi _1^{\ast } T_{\mathcal {Y}/S} \oplus \pi _2^{\ast } T_{\mathcal {M}/S}$
and in particular
$$\begin{align*}T_{f/S} \cong \pi_1^{\ast} T_{\mathcal{Y}/S} \oplus f^{-1} T_{\mathcal{M}/S}. \end{align*}$$
A.2.1 Construction of
$\mathcal {B}^{\bullet }(\mathcal {E})$
The definition of
$\mathcal {B}^{-1}(\mathcal {E})$
is analogous to that of
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{-1}(\mathcal {E})$
via the subquotient (34). One starts once again from the short exact sequence (33) but pulls it back all the way to
$\mathcal {E} nd(\mathcal {E}) \hookrightarrow \mathcal {D}^{(1)}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
and then pushes out along the trace
Similarly, we define
$\mathcal {B}^0(\mathcal {E})$
via the pull-back of the symbol exact sequence of
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{0}(\mathcal {E}) = \mathcal {A}_{f/S}(\mathcal {E})$
under the inclusion
$f^{-1} T_{\mathcal {M}/S} \hookrightarrow T_{f/S}$
arising through the splitting condition on equation (35) so that we have the following diagram:
Hence,
$\mathcal {B}^{\bullet }(\mathcal {E})$
is a subcomplex of
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{{\bullet }}(\mathcal {E})$
, and the following holds true.
Proposition A.2.2 [Reference Bloch and Esnault9, Sect. 5.2]
If the short exact sequence (35) is split, the complex
$\mathcal {B}^{\bullet }(\mathcal {E})$
is quasi-isomorphic to
${}^{\operatorname {tr}\!\!}{\mathcal {A}}^{{\bullet }}(\mathcal {E})$
.
Corollary A.2.3. The short exact sequence of complexes (31) is quasi-isomorphic to
Moreover, since we are considering only
$0^{th}$
direct images, we can drop the degree
$-2$
part of the first two complexes. Hence, we obtain a short exact sequence of complexes,
$$ \begin{align*}0 \to K_{\mathcal{X}/\mathcal{M}}[1] \to \mathcal{B}^{\bullet}(\mathcal{E}) \to \mathcal{C}^{\bullet}(\mathcal{E}) \to 0,\end{align*} $$
where
$\mathcal {C}^{-1}(\mathcal {E}) := \mathcal {E} nd(\mathcal {E})$
and
$\mathcal {C}^0(\mathcal {E}):= \mathcal {B}^0(\mathcal {E})$
. We also observe that
$\mathcal {C}^{\bullet }(\mathcal {E})$
is quasi-isomorphic to
$f^{-1}T_{\mathcal {M}/S}$
since this is exactly the cokernel of
$\mathcal {E} nd (\mathcal {E}) \to \mathcal {B}^0(\mathcal {E})$
. Thus, Theorem A.1.1 now simplifies to
Theorem A.2.4. We have an isomorphism of short exact sequences
Remark A.2.5. We observe that both sides of the central vertical isomorphism depend on
$\mathcal {E}$
.
A.2.2 Traceless version
${}^{0}{\mathcal {B}}^{\bullet }(\mathcal {E})$
of
$\mathcal {B}^{\bullet }(\mathcal {E})$
As expected, we define the subsheaf
${}^{0}{\mathcal {B}}^{-1}(\mathcal {E})\subset \mathcal {B}^{-1}(\mathcal {E})$
via the pull-back of the short exact sequence defining
$\mathcal {B}^{-1}(\mathcal {E})$
in equation (36) along the inclusion of traceless endomorphisms
$\mathcal {E} nd^0(\mathcal {E}) \hookrightarrow \mathcal {E} nd(\mathcal {E})$
,
As we did before, we introduce also a quotient sheaf
${}^{0}{\mathcal {B}}^{0}(\mathcal {E})$
of
$\mathcal {B}^0(\mathcal {E})$
, obtained as push-out through
$\mathcal {E} nd(\mathcal {E}) \rightarrow \mathcal {E} nd^0(\mathcal {E})$
, that is,
A.2.3 Identification of
$\mathcal {B}^{-1}(\mathcal {E})$
and
${}^{0}{\mathcal {B}}^{-1}(\mathcal {E})$
The duality
$$ \begin{align*}\mathcal{A}_{\mathcal{X}/\mathcal{M}}(\mathcal{E})^{\ast} \cong \mathcal{B}^{-1}(\mathcal{E})\end{align*} $$
was already stated in [Reference Bloch and Esnault9] formula (5.31). We give a proof here, in particular to include a discussion of the traceless case and to control the necessary restrictions on the characteristic of the ground field.
Theorem A.2.6. There is a canonical identification between the natural short exact sequences
There is also a traceless analogue:
Remark A.2.7. Note that the vertical maps on the right-hand side are given by the opposite of the isomorphism induced by the trace pairing.
Proof. Following [Reference Beĭlinson and Schechtman16, Sect. 2.1.1.3], let us define a pairing
$$ \begin{align*} \mathcal{E} \boxtimes \mathcal{E}^{\circ} (2\Delta) \times \mathcal{E} \boxtimes \mathcal{E}^{\circ} (\Delta) & \to \mathcal{O}_{\mathcal{X}};\\ (\psi_1,\psi_2) & \mapsto \widetilde{\operatorname{\mathrm{res}}}(\psi_1\cdot^t\psi_2), \end{align*} $$
where
denotes the transposition of
$\psi _2$
, that is the pull-back under the map that exchanges the two factors of the fibred product
$\mathcal {X} \times _{\mathcal {M}} \mathcal {X}$
. This means that
is a section of
$\mathcal {E}^{\circ } \boxtimes \mathcal {E}(\Delta )$
. Then we observe that the product
$\psi _1\cdot ^t\psi _2$
is a section of
$K_{\mathcal {X}/\mathcal {M}}\boxtimes K_{\mathcal {X}/\mathcal {M}}(3\Delta )$
, after taking the trace
$\operatorname {Tr} : \mathcal {E} \otimes \mathcal {E}^{\circ } \to K_{\mathcal {X}/\mathcal {M}}$
on each factor. Since
$\widetilde {\operatorname {\mathrm {res}}}$
is zero on
$K_{\mathcal {X}/\mathcal {M}}\boxtimes K_{\mathcal {X}/\mathcal {M}}(\Delta )$
, the pairing descends to a pairing on the quotients
$$\begin{align*}\langle - , - \rangle : \frac{\mathcal{E}\boxtimes \mathcal{E}^{\circ} (2\Delta)}{\mathcal{E}\boxtimes \mathcal{E}^{\circ}} \times \frac{\mathcal{E} \boxtimes \mathcal{E}^{\circ} (\Delta)}{\mathcal{E} \boxtimes \mathcal{E}^{\circ} (-\Delta)} \to \mathcal{O}_{\mathcal{X}}. \end{align*}$$
We claim that this pairing is nondegenerate. In order to check this, observe that it is defined on the central terms of the two short exact sequences (32) and (33),
Using the fact that
$\widetilde {\operatorname {\mathrm {res}}}$
vanishes on
$K_{\mathcal {X}/\mathcal {M}}\boxtimes K_{\mathcal {X}/\mathcal {M}}(\Delta )$
, we note that the pairing is identically zero when restricted to the product of the kernels
$\frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ } (\Delta )}{\mathcal {E}\boxtimes \mathcal {E}^{\circ }} \times \frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ }}{\mathcal {E} \boxtimes \mathcal {E}^{\circ }(-\Delta )}$
. Therefore, it induces pairings on the products of the kernel of one sequence with the quotient of the other one, that is, on
$\mathcal {E} nd(\mathcal {E}) \times \mathcal {E} nd(\mathcal {E})$
and
$\mathcal {E} nd(\mathcal {E}) \otimes T_{\mathcal {X}/\mathcal {M}} \times \mathcal {E} nd(\mathcal {E})\otimes K_{\mathcal {X}/\mathcal {M}}$
.
Lemma A.2.8. The residue pairing
$\langle - , - \rangle $
factorizes through the trace pairings
$- \operatorname {Tr}$
on
$\mathcal {E} nd(\mathcal {E}) \times \mathcal {E} nd(\mathcal {E})$
and
$+ \operatorname {Tr}$
on
$\mathcal {E} nd(\mathcal {E}) \otimes T_{\mathcal {X}/\mathcal {M}} \times \mathcal {E} nd(\mathcal {E})\otimes K_{\mathcal {X}/\mathcal {M}}$
.
Proof. Consider
$\psi _1$
a local section of
$\frac {\mathcal {E}\boxtimes \mathcal {E}^{\circ } (\Delta )} {\mathcal {E}\boxtimes \mathcal {E}^{\circ }} \subset \frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ } (2\Delta )}{\mathcal {E} \boxtimes \mathcal {E}^{\circ }}$
and
$\psi _2$
a local section of
$\frac {\mathcal {E}\boxtimes \mathcal {E}^{\circ } (\Delta )} {\mathcal {E}\boxtimes \mathcal {E}^{\circ }(-\Delta )}$
. As explained above,
$\langle \psi _1, \psi _2 \rangle $
depends only on
$\langle \psi _1, \overline {\psi _2} \rangle $
, where
$\overline {\psi _2}$
is the class of
$\psi _2$
in
$\frac {\mathcal {E}\boxtimes \mathcal {E}^{\circ } (\Delta )} {\mathcal {E}\boxtimes \mathcal {E}^{\circ }}$
. It will be enough to do the computations locally. Choose (as in [Reference Esnault and Tsai20]) a local coordinate x at a point
$p \in \mathcal {X}$
, and let
$(x,y)$
be the induced local coordinate at the point
$(p,p) \in \Delta $
. Then the local equation of
$\Delta $
is
$x-y = 0$
. Let
$e_i$
be a local basis of
$\mathcal {E}$
and
$e_j^*$
its dual basis. Then we can write the local sections
$\psi _1$
and
$\overline {\psi _2}$
as
$$ \begin{align*}\psi_1 = \sum_{i,j} e_i \otimes e_j^* \frac{\alpha_{ij}(x,y-x)} {y-x}dy \ \ \text{and} \ \ \overline{\psi_2} = \sum_{k,l} e_k \otimes e_l^* \frac{\beta_{kl}(x,y-x)} {y-x}dy\end{align*} $$
for some local regular functions
$\alpha _{ij}$
and
$\beta _{kl}$
. Then the local sections
$\phi _1$
and
$\phi _2$
of
$\mathcal {E} nd(\mathcal {E})$
associated to
$\psi _1$
and
$\overline {\psi _2}$
are given by
$$ \begin{align*}\phi_1 = \sum_{i,j} e_i \otimes e_j^* \alpha_{ij}(x,0) \ \ \text{and} \ \ \phi_2 = \sum_{k,l} e_k \otimes e_l^* \beta_{kl}(x,0). \end{align*} $$
Then we compute
$$ \begin{align*} \langle \psi_1, \overline{\psi_2} \rangle & = \widetilde{\operatorname{\mathrm{res}}}\left( \sum_{ijkl} e_i \otimes e_l^* \cdot e_k \otimes e_j^* \frac{\alpha_{ij}(x,y-x) \beta_{kl}(y,x-y)}{-(x-y)^2}dxdy\right) \\ & = \widetilde{\operatorname{\mathrm{res}}}\left( \sum_{ij} \frac{\alpha_{ij}(x,y-x) \beta_{ji}(y,x-y)}{-(x-y)^2} dxdy \right) \\ & = - \sum_{ij} \alpha_{ij}(x,0) \beta_{ji}(x,0) = - \operatorname{Tr}(\phi_1 \phi_2). \end{align*} $$
The computations for the second case are similar.
Since the trace pairing
$ \operatorname {Tr}$
is nondegenerate, we deduce from the above lemma that the pairing
$\langle - , - \rangle $
is also nondegenerate.
Now, we observe that
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E}) \subset \frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ }(2\Delta )}{\mathcal {E} \boxtimes \mathcal {E}^{\circ }}$
and that
$\frac {\mathcal {E} \boxtimes \mathcal {E}^{\circ }(\Delta )}{\mathcal {E} \boxtimes \mathcal {E}^{\circ }(-\Delta )}\twoheadrightarrow \mathcal {B}^{-1}(\mathcal {E})$
. We want to prove that the restriction
$\langle \mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E}), - \rangle $
descends to
$\mathcal {B}^{-1}(\mathcal {E})$
, but this follows from the definition of
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
by pull-back via
$T_{\mathcal {X}/\mathcal {M}}\otimes \mathcal {E} nd(\mathcal {E})$
and the definition of
$\mathcal {B}^{-1}(\mathcal {E})$
by push-out via
$\mathcal {E} nd(\mathcal {E})\otimes K_{\mathcal {X}/\mathcal {M}} \stackrel {Tr}{\twoheadrightarrow } K_{\mathcal {X}/\mathcal {M}}$
, and the duality between these two maps. Hence, we obtain a nondegenerate pairing
$$\begin{align*}\langle - , - \rangle: \mathcal{A}_{\mathcal{X}/\mathcal{M}}(\mathcal{E}) \times \mathcal{B}^{-1}(\mathcal{E}) \longrightarrow \mathcal{O}_{\mathcal{X}}. \end{align*}$$
The same argument yields nondegeneracy of the traceless version of this pairing:
$$\begin{align*}\langle - , - \rangle: \mathcal{A}_{\mathcal{X}/\mathcal{M}}^0(\mathcal{E}) \times {}^{0}{\mathcal{B}}^{-1}(\mathcal{E}) \longrightarrow \mathcal{O}_{\mathcal{X}}.\\[-32pt] \end{align*}$$
Remark A.2.9. The duality between
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
and
$\mathcal {B}^{-1}(\mathcal {E})$
was constructed by Sun–Tsai in [Reference Sun and Tsai50, Lemma 4.11.2] using a local description of
$\mathcal {B}^{-1}(\mathcal {E})$
. Note that their claim involves the Atiyah algebroid
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E}^*)$
, which is isomorphic to
$\mathcal {A}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})$
but has opposite extension class.
Remark A.2.10. We note that
$$ \begin{align*}\frac{\mathcal{E}\boxtimes \mathcal{E}^{\circ}(\Delta)}{\mathcal{E}\boxtimes \mathcal{E}^{\circ}(-\Delta)}\cong \mathcal{D}^{(1)}_{\mathcal{X}/\mathcal{M}}(\mathcal{E}) \otimes K_{\mathcal{X}/\mathcal{M}}.\end{align*} $$
Thus, the pairing
$\langle -,- \rangle $
described in the above proof induces a natural isomorphism between
$\mathcal {D}^{(1)}_{\mathcal {X}/\mathcal {M}}(\mathcal {E})^*$
and
$\mathcal {D}^{(1)}_{\mathcal {X}/\mathcal {M}}(\mathcal {E}) \otimes K_{\mathcal {X}/\mathcal {M}}$
.
Appendix B The splitting of the adjoint map
In this appendix, we collect some representation-theoretical facts needed in the proof of Proposition 5.0.2. We will work in the following framework. We will denote by
$\mathcal {E}$
a rank r vector bundle on a smooth algebraic variety X and as usual
$\mathcal {E}nd^0(\mathcal {E})$
will denote the traceless endomorphisms of
$\mathcal {E}$
. We need the characteristic p of the field
$\Bbbk $
to be
$0$
or not dividing r.
First, we observe that we have two nondegenerate pairings induced by the trace,
$$ \begin{align} \operatorname{Tr} : \mathcal{E} nd(\mathcal{E})\times \mathcal{E} nd(\mathcal{E}) & \to \mathcal{O}_X , \end{align} $$
$$ \begin{align} \operatorname{Tr} : \mathcal{E} nd(\mathcal{E} nd(\mathcal{E})) \times \mathcal{E} nd(\mathcal{E} nd(\mathcal{E})) & \to \mathcal{O}_X , \end{align} $$
which allow us to identify
$\mathcal {E} nd (\mathcal {E})$
with
$\mathcal {E} nd(\mathcal {E})^*$
and
$\mathcal {E} nd(\mathcal {E} nd(\mathcal {E}))$
with
$\mathcal {E} nd(\mathcal {E} nd(\mathcal {E}))^*$
. Moreover, we denote by
$$ \begin{align} \begin{aligned} \mathrm{ad}: \mathcal{E} nd(\mathcal{E}) & \to \mathcal{E} nd(\mathcal{E} nd(\mathcal{E})) \\ \alpha & \mapsto (\beta \mapsto [\alpha, \beta]) \end{aligned} \end{align} $$
the
$\mathcal {O}_X$
-linear map given by the adjoint for any local sections
$\alpha , \beta $
of
$\mathcal {E} nd(\mathcal {E})$
.
Lemma B.0.1. Let
$\alpha ,\beta $
be local sections of the vector bundle
$\mathcal {E} nd(\mathcal {E})$
. The
$\mathcal {O}_X$
-linear map
$$ \begin{align*} s:\mathcal{E} nd(\mathcal{E} nd(\mathcal{E})) \cong \mathcal{E} nd (\mathcal{E})\otimes \mathcal{E} nd(\mathcal{E}) & \to \mathcal{E} nd(\mathcal{E})\\ \alpha \otimes \beta & \mapsto \frac{1}{2r}[\beta,\alpha] \end{align*} $$
satisfies
$s \circ \mathrm {ad}(\alpha ) = \alpha - \frac {\operatorname {\mathrm {tr}}(\alpha )}{r}\operatorname {Id}_{\mathcal {E}}$
, i.e., s is a splitting of the restriction of
$\mathrm {ad}$
to
$\mathcal {E} nd^0(\mathcal {E})$
.
Proof. It will be enough to check the equality pointwise. The statement then reduces to check that for an
$r \times r$
matrix
$A \in \mathrm {M}_r(\Bbbk )$
we have the equality
$s \circ \mathrm {ad} (A) = A - \frac {\operatorname {\mathrm {tr}}(A)}{r} I_r$
. We consider the canonical basis
$\{ E_{ij} \}$
with
$1 \leq i,j \leq r$
of
$\mathrm {M}_r(\Bbbk )$
. The dual basis of
$\{ E_{ij} \}$
under the trace pairing (37) is given by
$\{ E_{ji} \}$
. The claim then follows by straightforward computation:
$$ \begin{align*} s \circ \mathrm{ad}(A) & = s \left( \sum_{i,j} E_{ji} \otimes [A,E_{ij}] \right) = \frac{1}{2r} \sum_{i,j} AE_{ij}E_{ji} - E_{ij}AE_{ji} - E_{ji}AE_{ij} + E_{ji}E_{ij}A \\ & = \frac{1}{2r} (2r A - 2\operatorname{\mathrm{tr}}(A) I_r).\\[-36pt] \end{align*} $$
Lemma B.0.2. Using the identifications (37) and (38) given by the trace pairings, we denote by
$s^* : \mathcal {E} nd(\mathcal {E}) \to \mathcal {E} nd(\mathcal {E} nd(\mathcal {E}))$
the dual of s. Then we have the equality
$$ \begin{align*}s^* = \frac{1}{2r} \mathrm{ad}.\end{align*} $$
Proof. As in the previous lemma, we will check the equality pointwise. By the definition of the dual map
$s^*$
and the trace pairings (37) and (38), it is easily seen that the claimed equality is equivalent to the equality
$$ \begin{align*}\operatorname{\mathrm{tr}} (\mathrm{ad}(A). B \otimes C) = \operatorname{\mathrm{tr}} (A [C,B])\end{align*} $$
for any matrices
$A,B,C \in \mathrm {M}_r(\Bbbk )$
. Note that the trace on the left-hand side is the trace on
$End(\mathrm {M}_r(\Bbbk )) \cong \mathrm {M}_r(\Bbbk ) \otimes \mathrm {M}_r(\Bbbk )$
. Again, this equality is proved by straightforward computation:
$$ \begin{align*} \operatorname{\mathrm{tr}}(\mathrm{ad}(A). B \otimes C) & = \sum_{i,j} \operatorname{\mathrm{tr}} ( E_{ji} \otimes [A,E_{ij}] \otimes B \otimes C ) = \sum_{i,j} \operatorname{\mathrm{tr}}(E_{ji} C ) \operatorname{\mathrm{tr}}( [A,E_{ij}] B) \\ & = \sum_{i,j} (\operatorname{\mathrm{tr}}(E_{ji} C ) ( \operatorname{\mathrm{tr}} (BAE_{ij}) - \operatorname{\mathrm{tr}} (E_{ij}AB) ) = \operatorname{\mathrm{tr}}(BAC) - \operatorname{\mathrm{tr}}(ABC) \\ & = \operatorname{\mathrm{tr}}(A[C,B]).\\[-36pt] \end{align*} $$
We will also abuse slightly of notation and denote also by
$\mathrm {ad}$
the
$\mathcal {O}_X$
-linear map
$\mathcal {E} nd(\mathcal {E}) \to \mathcal {E} nd(\mathcal {E} nd^0(\mathcal {E}))$
induced by the one defined in equation (39). We will write instead
$\mathrm {ad}_0:\mathcal {E} nd^0(\mathcal {E}) \to \mathcal {E} nd^0(\mathcal {E} nd^0(\mathcal {E}))$
for the restriction to
$\mathcal {E} nd^0(\mathcal {E})$
.
Proposition B.0.3.
-
(a) There exists a
$\mathcal {O}_X$
-linear map extending, respectively,
$$ \begin{align*}\widetilde{\mathrm{ad}}: \mathcal{A}(\mathcal{E}) \to \mathcal{A}(\mathcal{E} nd^0(\mathcal{E})), \end{align*} $$
$\mathrm {ad}$
inducing the identity on
$T_X$
. Note that
$\widetilde {\mathrm {ad}}$
factorizes through
$\mathcal {A}^0(\mathcal {E})$
. We shall denote by the factorized map.
$$ \begin{align*}\widetilde{\mathrm{ad}}_0 : \mathcal{A}^0(\mathcal{E}) \to \mathcal{A}(\mathcal{E} nd^0(\mathcal{E})) \end{align*} $$
-
(b) There exists a
$\mathcal {O}_X$
-linear map extending
$$ \begin{align*}\widetilde{s}: \mathcal{A}(\mathcal{E} nd^0(\mathcal{E})) \to \mathcal{A}^0(\mathcal{E}), \end{align*} $$
$s: \mathcal {E} nd(\mathcal {E} nd^0(\mathcal {E})) \to \mathcal {E} nd^0(\mathcal {E})$
, inducing the identity on
$T_X$
and such that
$\widetilde {s} \circ \widetilde {\mathrm {ad}}_0 = Id_{\mathcal {A}^0(\mathcal {E})}$
.
-
(c) With the notation of Appendix A, there exists a
$\mathcal {O}_{\mathcal {X}}$
-linear map lifting
$$ \begin{align*}\widehat{\mathrm{ad}}:{}^0\mathcal{B}^{-1}(\mathcal{E}) \to{}^0\mathcal{B}^{-1}(\mathcal{E} nd^0(\mathcal{E})),\end{align*} $$
$\mathrm {ad}_0$
and inducing
$2r \operatorname {Id}$
on the line subbundle
$K_{\mathcal {X}/\mathcal {M}}$
.
Proof. Part (a) is proved in [Reference Atiyah5] pp. 188–189.
Part (b): We define
$\widetilde {s}$
as the push-out of the exact sequence
$$ \begin{align*}0 \to \mathcal{E} nd^0 (\mathcal{E} nd^0(\mathcal{E})) \to \mathcal{A}^0(\mathcal{E} nd^0(\mathcal{E})) \to T_X \to 0\end{align*} $$
under the
$\mathcal {O}_X$
-linear map s. Then, by Lemma B.0.1, since s is a splitting of
$\mathrm {ad}_0$
, we see that the extension class of the push-out is the same as the extension class of
$\mathcal {A}^0(\mathcal {E})$
; hence, these two vector bundles are isomorphic (see, e.g., [Reference Atiyah5] pp. 188–189).
Part (c): We recall from Theorem A.2.6 that there exist isomorphisms
$$ \begin{align*}\delta_{\mathcal{E}} : \mathcal{A}^0_{\mathcal{X}/\mathcal{M}}(\mathcal{E})^* \to {}^0\mathcal{B}^{-1}(\mathcal{E}) \ \ \text{and} \ \ \delta_{\mathcal{E} nd^0(\mathcal{E})} : \mathcal{A}^0_{\mathcal{X}/\mathcal{M}}(\mathcal{E} nd^0(\mathcal{E}))^* \to {}^0\mathcal{B}^{-1}(\mathcal{E} nd^0(\mathcal{E})).\end{align*} $$
We then construct the map
$\widehat {\mathrm {ad}}$
as the composition
$$ \begin{align*}\widehat{\mathrm{ad}} = (2r) \delta_{\mathcal{E} nd^0(\mathcal{E})} \circ \widetilde{s}^* \circ \delta_{\mathcal{E}}^{-1}.\end{align*} $$
Then
$\widehat {\mathrm {ad}}$
induces
$(2r) \operatorname {Id}$
on
$K_{\mathcal {X}/\mathcal {M}}$
and, by Lemma B.0.2,
$\widehat {\mathrm {ad}}$
lifts the map
$\mathrm {ad}_0$
.
Remark B.0.4. Proposition B.0.3 coincides with [Reference Sun and Tsai50, Prop. 3.10]. Our proof is different since we give a global construction of the liftings of the adjoint maps.
Appendix C Basic facts about the moduli space
$\mathcal {M}$
through the Hitchin system
In this appendix, we give proofs for some of the basic facts about the moduli space of stable bundles
$\mathcal {M}$
(as in Section 4.1) that we use in the main body of the paper. These are essentially all well known, but we were unable to find references for them in the generality we need (outside the complex case). We therefore show here how they can all be obtained using the Hitchin system—a strategy once again due to Hitchin (cfr. [Reference Hitchin29, §6] and [Reference Hitchin30, §5])—via some minor adaptations to the algebro-geometric setting.
C.1 The moduli space of Higgs bundles and the Hitchin system
We will denote by
$\mathcal {M}^{\operatorname {H}, \operatorname {ss}}$
the moduli space of semi-stable Higgs bundles with trivial determinant (and trace-free Higgs field)—all still relative over S as before. This space is singular but normal and comes equipped with the Hitchin system, a projective morphism
$\phi $
to the vector bundle
$\pi _{\mathcal {H}}:\mathcal {H}\rightarrow S$
associated to the sheaf
$\oplus _{i=2}^r \pi _{s *} K_{\mathcal {C}/S}^i$
over S. This morphism is equivariant with respect to the
$\mathbb {G}_m$
-action that scales the Higgs fields and acts with weight i on
$\pi _{s *} K^i_{\mathcal {C}/S}$
. The fibres of
$\pi _{\operatorname {H}}:\mathcal {M}^{\operatorname {H}, \operatorname {ss}}\rightarrow S$
have a canonical (algebraic) symplectic structure on their smooth locus, which extends the one on
$T^*_{\mathcal {M}/S}$
. Closed points in
$\mathcal {H}$
give rise to degree r spectral covers of
$\mathcal {C}$
. The locus whose spectral curve is smooth is denoted by
$\mathcal {H}^{\operatorname {reg}}$
.
C.2 Proofs
Proposition C.2.1 Proposition 4.1.1(c)
There are no global vector fields on
$\mathcal {M}$
:
$$ \begin{align*}\pi_{e*}T_{\mathcal{M}/S}=\{0\}.\end{align*} $$
Proof. Elements of
$\pi _{e*}T_{\mathcal {M}/S}$
would give rise to global functions on
$T^*_{\mathcal {M}/S}$
. As the complement of
$\mathcal {M}$
in
$\mathcal {M}^{\operatorname {H}, \operatorname {ss}}$
has sufficiently high codimension, these would extend by Hartogs’s theorem to all of
$\mathcal {M}^{\operatorname {H}, \operatorname {ss}}$
. As they have weight
$1$
under the
$\mathbb {G}_m$
-action, they have to be pulled back from functions on
$\mathcal {H}$
of the same weight, but there are no such functions.
Proposition C.2.2. The Hitchin symbol
$\rho ^{\operatorname {Hit}}$
is an isomorphism.
Proof. Elements of
${\pi _e}_{\ast } \operatorname {\mathrm {Sym}}^2 T_{\mathcal {M} / S}$
can be understood as regular functions on the total space of
$T^*_{\mathcal {M}/S}$
, of degree
$2$
on all tangent spaces. In turn, these extend, by Hartogs’s theorem, to
$\mathcal {M}^{\operatorname {H},\operatorname {ss}}$
, where they are of degree 2 with respect to the
$\mathbb {G}_m$
-action that scales the Higgs field. As the Hitchin system is equivariant, they are moreover obtained from regular linear functions on the quadratic part of the Hitchin base, which is exactly given by
$R^1 {\pi _s}_{\ast } T_{\mathcal {C} / S}$
though
$\rho ^{\operatorname {Hit}}$
.
To establish that
$\mu _{\mathcal {L}^k}$
is injective, we can again adapt the reasoning from [Reference Hitchin30, §5]. By Propositions 3.6.1 and 4.7.1 and Lemma C.2.2, it suffices to show that
$\Phi $
is injective.
Lemma C.2.3 [Reference Hitchin30, Proposition 5.2]
There exists a canonical isomorphism
of
$\pi _{\mathcal {H}*}\mathcal {\mathcal {O}_{\mathcal {H}}}$
-modules which is equivariant with respect to the natural action of
$\mathbb {G}_m$
on
$\pi _{\mathcal {H}*}\mathcal {\mathcal {O}_{\mathcal {H}}}\otimes \mathcal {H}$
and the natural action twisted by weight
$-1$
on
$R^1\pi _{\operatorname {H} *}\mathcal {O}$
.
Proof. Indeed, sections of
$\mathcal {H}^*$
give rise to fibre-wise linear functions on
$\mathcal {H}$
, which pull back by
$\phi $
to functions on
$\mathcal {M}^{\operatorname {H}}$
. As the latter has an algebraic symplectic structure on
$\mathcal {M}^{\operatorname {H}, \operatorname {s}}$
extending the canonical one on
$T^*{\mathcal {M}}$
, these give rise to Hamiltonian vector fields on
$\mathcal {M}^{\operatorname {H}, \operatorname {s}}$
which are tangent to the fibres of
$\phi $
. Moreover, the inverse of the determinant-of-cohomology line bundle
$\mathcal {L}$
naturally extends to
$\mathcal {M}^{\operatorname {H}}$
and is relatively ample with respect to
$\phi $
. Taking the cup product with its relative Atiyah class gives a natural morphism
$\pi _{\operatorname {H} *} T_{\mathcal {M}^{\operatorname {H}}/S}\rightarrow R^1\pi _{\operatorname {H} *}\mathcal {O}$
. The composition gives a morphism
$\mathcal {H}\rightarrow R^1\pi _{\operatorname {H} *}\mathcal {O}$
, which naturally extends as a morphism of
$\pi _{\mathcal {H}*}\mathcal {\mathcal {O}_{\mathcal {H}}}$
-modules to the desired morphism
$\Psi $
.
To show that
$\Psi $
is an isomorphism, it can be argued as follows: As
$\pi _{\operatorname {H}}$
factors over
$\pi _{\mathcal {H}}$
, and the latter is an affine morphism, we have that
$R^1\pi _{\operatorname {H} *}\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}\cong \pi _{\mathcal {H}*}\left (R^1\phi _{*}\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}\right )$
. Now, through the theory of abelianisation, we know that over a locus
$\mathcal {H}^{\circ }$
whose complement has sufficiently high co-dimension, the morphism
$\phi $
is a family of (semi-)abelian varieties. The line bundle
$\mathcal {L}$
restricts to an ample one on the fibres, and for those fibres X it is known that cupping with
$[\mathcal {L}]$
is an isomorphism
$H^0(X, T_X)\rightarrow H^1(X, \mathcal {O}_X)$
. As the vector fields on
$\mathcal {M}^{\operatorname {H}}$
are independent, on each such X the space
$H^0(X, T_X)$
is given by the vector field coming from
$\mathcal {H}^*$
. As a result, we find that, on
$\mathcal {H}^{\circ }$
,
$R^1\phi _{*}\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}$
is a trivial vector bundle and that the map
$\Psi $
is indeed an isomorphism.
It is also straightforward to observe that the map
$\Psi $
is in fact equivariant for the natural
$\mathbb {G}_m$
-action that is defined on all spaces, induced by the scaling of Higgs fields, provided that we twist the action on
$R^1\pi _{\mathcal {H}*}\mathcal {M}^{\operatorname {H}}$
by a weight
$-1$
.
Proof. It suffices to remark that sections of
$R^1\pi _{e*}\mathcal {O}_{\mathcal {M}}$
correspond to sections of
$R^1\pi _{\operatorname {H}*}\mathcal {O}_{\mathcal {M}^{\operatorname {H},\operatorname {ss}}}$
of weight
$0$
, which would correspond under
$\Psi $
to sections of weight
$-1$
, of which there are none.
Proposition C.2.5. The map
$\cup [\mathcal {L}]:\pi _{e*}\operatorname {\mathrm {Sym}}^2T_{\mathcal {M}/S}\rightarrow R^1\pi _{e*} T_{\mathcal {M}/S}$
is an isomorphism.
Proof. We now want to restrict the isomorphism
$\Psi $
from C.2.3 to the subbundle of
$\pi _{\mathcal {H}*}\mathcal {O}_{\mathcal {H}}\otimes \mathcal {H}^*$
of weight 2, which corresponds exactly to fibre-wise linear functionals on
$\pi _{s*}K^2_{\mathcal {C}/S}$
, which by relative Serre duality is exactly given by
$R^1\pi _{s*} T_{\mathcal {C}/S}$
. On this space,
$\Psi $
restricts to give an isomorphism to
$R^1\pi _{e*}T_{\mathcal {M}/S}$
. To show that this is a multiple of
$\Phi $
, one can argue as follows: If
$\mathcal {O}^{(1)}$
is the structure sheaf of the first-order infinitesimal neighborhood of
$\mathcal {M}$
in
$\mathcal {M}^{\operatorname {H}}$
(cfr. [53, Tag 05YW]), we have the short exact sequence on
$\mathcal {M}$
Here,
$N^*_{\mathcal {M}/\mathcal {M}^{\operatorname {H}}}$
is the co-normal bundle of
$\mathcal {M}$
in
$\mathcal {M}^{\operatorname {H}}$
, which is canonically isomorphic to the tangent bundle
$T_{\mathcal {M}/S}$
. As by Proposition C.2.4 we have that
$R^1\pi _{e*}\mathcal {O}_{\mathcal {M}}=\{0\}$
, this gives
$$ \begin{align*} R^1\pi_{e*} T_{\mathcal{M}/S}\cong R^1\pi_{e*}\mathcal{O}^{(1)}. \end{align*} $$
If
$\mathcal {I}$
is the ideal sheaf of
$\mathcal {M}$
in
$\mathcal {M}^{\operatorname {H}}$
, we have that
$\mathcal {O}^{(1)}=\left (\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}\big / \mathcal {I}^2\right ) \Big |_{\mathcal {M}}$
, and hence, we have a restriction map
which is the identity on
$R^1\pi _{e*} T_{\mathcal {M}/S}$
(sitting inside
$R^1\pi _{\operatorname {H} *}\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}$
as the weight
$1$
part). So we only need to keep track of first-order information in the normal direction. We now claim that, for any
$\Delta \in R^1\pi _{\operatorname {H} *}(\Omega ^1_{\mathcal {M}^{\operatorname {H}}/S})$
which restricts to
$\widetilde {\Delta }\in R^1\pi _{e*}(\Omega ^1_{\mathcal {M}/S})$
, the following diagram is commutative:
In [Reference Hitchin30, page 379], this was shown using holomorphic Darboux coordinates on the total space of
$T_{\mathcal {M}/S}$
, coming from (holomorphic) coordinates on
$\mathcal {M}$
. The reasoning does not strictly speaking need the latter choice though, and it suffices to work with a local trivialisation of
$T_{\mathcal {M}/S}$
. In this sense, it also goes through in an algebraic context, as follows. Let
$U_{i}$
be a covering of
$\mathcal {M}$
by open affines, such that
$T_{\mathcal {M}/S}\big |_{U_{i}}$
is free. For a fixed i, we choose generators
$e_1, \ldots , e_n$
of the latter. These can also be understood as functions
$f_1, \ldots , f_n$
on
$T^*_{\mathcal {M}/S}\big |_{U_{\gamma }}$
. If we denote the dual sections to
$e_1, \ldots , e_n$
as
$e^1, \ldots , e^n$
, then we can interpret their pull backs as one-forms on the total space of
$T^*_{\mathcal {M}/S}\big |_{U_{\gamma }}$
. The tautological one-form
$\theta $
on the total space of
$T^*_{\mathcal {M}/S}$
can now be written locally as
$\theta =\sum _{\alpha } f_{\alpha } e^{\alpha }$
, and the canonical symplectic form is therefore
$\omega =-d\theta =\sum _{\alpha } df_{\alpha }\wedge e^{\alpha }$
. If a section of
$\pi _{e*}\operatorname {\mathrm {Sym}}^2 T_{\mathcal {M}/S}$
is locally written as
$G=\sum _{\alpha ,\beta }G^{\alpha \beta }e_{\alpha }\odot e_{\beta }$
(with the
$G^{\alpha \beta } \in \mathcal {O}_{U_{i}}$
), then the corresponding element of
$\pi _{\operatorname {H}*}\mathcal {O}_{\mathcal {M}^{\operatorname {H}}}$
can be written as
$\sum _{\alpha ,\beta }G^{\alpha \beta }f_{\alpha }f_{\beta }$
. The corresponding Hamiltonian vector field (with respect to
$\omega $
) in
$\pi _{\operatorname {H} *}$
is locally written as
$$ \begin{align*} -\sum_{\alpha,\beta,\gamma}e_{\gamma}(G^{\alpha\beta})f_{\alpha} f_{\beta}h^{\gamma}+2\sum_{\alpha,\beta}G^{\alpha\beta}f_{\alpha} e_{\beta}, \end{align*} $$
(where, with a slight abuse of notation, we denote by
$e_1, \ldots , e_n, h^1, \ldots , h^n$
the elements of the basis of
$T_{\mathcal {M}^{\operatorname {H}}/S}$
dual to
$e^1, \ldots , e^n, df_1, \ldots , df_n$
). After taking the cup product with
$\Delta $
(which we represent by a Čech cohomology class with respect to the open covering
$T^*_{U_{i}/S}$
) and restricting to
$\mathcal {O}^{(1)}$
, this gives indeed
$2G\cup \widetilde {\Delta }$
. We conclude by applying this to
$\Delta =[\mathcal {L}]$
, in which case the ‘bottom path’ of equation (40) is given by a component of the isomorphism
$\Psi $
.
Corollary C.2.6. The map
$\Phi $
from (15) is an isomorphism.
Proof. This follows immediately by combining Proposition C.2.2, Proposition C.2.5 and Proposition 4.7.1.
Finally, as a corollary we also get the final fact we need in the proof of the flatness of the Hitchin connection (Theorem 4.8.2):
Lemma C.2.7. The map
$\mu _{\mathcal {L}^k}$
is injective.
Supporting information
TB was supported in part by FCT/Portugal through the projects UID/MAT/04459/2013 and PTDC/MAT-GEO/3319/2014. JM was supported in part by EPSRC grant EP/N029828/1. CP was supported in part by the Marie Curie project GEOMODULI of the programme FP7/PEOPLE/2013/CIG, project number 618471.
Competing Interest
None.
