Isoholonomic relative connections

Let $\unicode[STIX]{x1D70C}\,:\,M\rightarrow S$ be a submersion of complex manifolds and let ${\mathcal{F}}$ be a holomorphic vector bundle on $M$ of rank $r$ . We need the following three notions relative to $\unicode[STIX]{x1D70C}$ .

Let us comment on these definitions. We begin with observing that if $\unicode[STIX]{x1D70C}$ factors through a submersion $\unicode[STIX]{x1D70C}^{\prime }\,:\,M\rightarrow S^{\prime }$ and one of the three properties above holds for $\unicode[STIX]{x1D70C}$ , then that property also holds for $\unicode[STIX]{x1D70C}^{\prime }$ .

Next we note that we can drop the adjective ‘locally’ in the definition of a $\unicode[STIX]{x1D70C}$ -basic form if the fibers of $\unicode[STIX]{x1D70C}$ are connected: it is then just the pull-back of a form on $S$ . This is still true if the set of $v\in S$ for which $\unicode[STIX]{x1D70C}^{-1}(v)$ is connected contains an open-dense subset of $S$ (we then say that a general fiber of $\unicode[STIX]{x1D70C}$ is connected).

For a flat $\unicode[STIX]{x1D70C}$ -connection $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ on ${\mathcal{F}}$ , its flat local sections make up a subsheaf $\mathbb{F}\subseteq {\mathcal{F}}$ . This is a locally free $\unicode[STIX]{x1D70C}^{-1}{\mathcal{O}}_{S}$ -submodule of rank $r$ with the property that the natural map ${\mathcal{O}}_{M}\otimes _{\unicode[STIX]{x1D70C}^{-1}{\mathcal{O}}_{S}}\mathbb{F}\rightarrow {\mathcal{F}}$ is an isomorphism. The converse also holds: a subsheaf $\mathbb{F}\subseteq {\mathcal{F}}$ with these properties determines a flat $\unicode[STIX]{x1D70C}$ -connection on ${\mathcal{F}}$ . This also amounts to giving a (maximal) atlas of local holomorphic trivializations of ${\mathcal{F}}$ whose transition functions factor through $\unicode[STIX]{x1D70C}$ . In the situations that we shall consider, $\unicode[STIX]{x1D70C}$ will be topologically locally trivial with connected fibers, and then a given flat $\unicode[STIX]{x1D70C}$ -connection is isoholonomic precisely if its holonomy along $\unicode[STIX]{x1D70C}^{-1}v$ (given as a $\operatorname{GL}(r,\mathbb{C})$ -orbit in $\operatorname{Hom}(\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D70C}^{-1}v),\operatorname{GL}(r,\mathbb{C}))$ ) is locally constant in $v\in S$ in an evident sense. Whence the terminology.

Let an isoholonomic $\unicode[STIX]{x1D70C}$ -connection $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ on ${\mathcal{F}}$ be given. We then have an open covering $\{V_{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}}$ of $S$ and for every $\unicode[STIX]{x1D6FC}$ a flat holomorphic connection $\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D6FC}}\,:\,{\mathcal{F}}|\unicode[STIX]{x1D70C}^{-1}V_{\unicode[STIX]{x1D6FC}}\rightarrow \unicode[STIX]{x1D6FA}_{M}\otimes {\mathcal{F}}|\unicode[STIX]{x1D70C}^{-1}V_{\unicode[STIX]{x1D6FC}}$ which lifts $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}|\,:\,\unicode[STIX]{x1D70C}^{-1}V_{\unicode[STIX]{x1D6FC}}\,:\,{\mathcal{F}}|\unicode[STIX]{x1D70C}^{-1}V_{\unicode[STIX]{x1D6FC}}\rightarrow \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70C}}\otimes {\mathcal{F}}|\unicode[STIX]{x1D70C}^{-1}V_{\unicode[STIX]{x1D6FC}}$ . If $\{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}}$ is a $C^{\infty }$ partition of unity on $S$ with $\sup (\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}})\subseteq V_{\unicode[STIX]{x1D6FC}}$ , then $\unicode[STIX]{x1D6FB}:=\sum _{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70C}^{\ast }(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}})\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D6FC}}$ is a $C^{\infty }$ -connection on ${\mathcal{F}}$ which globally lifts $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ in a particular way: in terms of a local trivialization in the atlas described above, this connection is given by a matrix of $\unicode[STIX]{x1D70C}$ -basic forms of type $(1,0)$ . Its curvature is therefore given by a matrix of $\unicode[STIX]{x1D70C}$ -basic 2-forms of Hodge level ${\geqslant}1$ (i.e., is a sum of forms of types $(2,0)$ and $(1,1)$ ). Hence, the Chern form $\operatorname{C}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB})$ is a $\unicode[STIX]{x1D70C}$ -basic $2k$ -form of Hodge level ${\geqslant}k$ . Note that this remains so if we alter the connection $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ by adding to it a nilpotent relative differential $\unicode[STIX]{x1D702}$ (i.e., a section of $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70C}}\otimes {\mathcal{E}}\!nd({\mathcal{F}})$ that takes values in the nilpotent endomorphisms), for then the curvature form of $\unicode[STIX]{x1D6FB}$ will be nilpotent along the fibers of $\unicode[STIX]{x1D70C}$ and so $\operatorname{C}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB}+\unicode[STIX]{x1D702})$ will map to zero in $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70C}}^{2k}$ . Since $\operatorname{C}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB}+\unicode[STIX]{x1D702})$ is also closed, it is then still $\unicode[STIX]{x1D70C}$ -basic. So, when the general fiber of $\unicode[STIX]{x1D70C}$ is connected, $\operatorname{C}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB}+\unicode[STIX]{x1D702})$ is the pull-back of one on $S$ . In particular, the complex $k$ th Chern class of ${\mathcal{F}}$ lies in the image of $\unicode[STIX]{x1D70C}^{\ast }\,:\,H^{2k}(S;\mathbb{C})\rightarrow H^{2k}(M;\mathbb{C})$ .

But, as we will see in our main application, it is possible for $\mathbb{F}$ to have nontrivial holonomy along the fibers of $\unicode[STIX]{x1D70C}$ and so $\mathbb{F}$ need not be a sheaf pull-back of a holomorphic vector bundle on  $S$ .

We next extend this to a stratified setting. This naturally leads us to consider ‘germ versions’ of the notions we just introduced. Let $X$ be a complex-analytic variety endowed with a stratification ${\mathcal{S}}$ , by which we mean a finite partition of $X$ into connected nonsingular locally closed subvarieties, called strata, such that the closure of a stratum is a subvariety that is a union of strata. We partially order the collection of strata by letting $S^{\prime }\leqslant S$ mean that $S^{\prime }\subseteq \overline{S}$ .

Here $X_{S}$ denotes the germ of $X$ at $S$ and so this means that $\unicode[STIX]{x1D70C}_{S}$ is represented by an analytic retraction whose domain is an unspecified neighborhood $U_{S}$ of $S$ in $X$ . If the stratification ${\mathcal{S}}$ satisfies Whitney’s $(a)$ condition, then we may take $U_{S}$ so small such that for every $S^{\prime }\in {\mathcal{S}}$ , $\unicode[STIX]{x1D70C}_{S}|U_{S}\cap S^{\prime }$ is a submersion. Note that for every $S\in {\mathcal{S}}$ , the collection $\{\unicode[STIX]{x1D70C}_{S^{\prime }}|\overline{S}\}_{S^{\prime }<S}$ is a system of retractions for $(\overline{S},{\mathcal{S}}|\overline{S})$ . A complex submanifold of a complex manifold need not be a holomorphic retract of some neighborhood of it and so the mere existence of such a system indicates that the stratification is quite special. A standard example is the natural stratification of a torus embedding. We will see that the Satake and toric compactifications of a locally symmetric variety also come with this structure.

We make the rigidified stratified varieties objects of a category: a morphism $(\tilde{X},\tilde{{\mathcal{S}}},\tilde{\unicode[STIX]{x1D70C}})\rightarrow (X,{\mathcal{S}},\unicode[STIX]{x1D70C})$ is given by a complex-analytic morphism $\unicode[STIX]{x1D70B}\,:\,\tilde{X}\rightarrow X$ that takes any stratum $\tilde{S}$ of $\tilde{{\mathcal{S}}}$ submersively to a stratum $S$ of ${\mathcal{S}}$ in such a manner that on a neighborhood of $\tilde{S}$ we have $\unicode[STIX]{x1D70B}\tilde{\unicode[STIX]{x1D70C}}_{\tilde{S}}=\unicode[STIX]{x1D70C}_{S}\unicode[STIX]{x1D70B}$ and we demand that the preimage of the union of the open strata in $X$ is equal to the union of the open strata in $\tilde{X}$ .

From now on $(X,{\mathcal{S}},\unicode[STIX]{x1D70C})$ is a rigidified stratified variety with $X$ topologically normal (in the sense that normalization is a homeomorphism). We denote by ${X\unicode[STIX]{x0030A}}\subseteq X$ the union of the open strata and by $j\,:\,{X\unicode[STIX]{x0030A}}\subseteq X$ and $i_{S}\,:\,S\subseteq X$ ( $S\in {\mathcal{S}}$ ) the inclusions. We write ${\unicode[STIX]{x1D70C}\unicode[STIX]{x0030A}}_{S}$ for the restriction of $\unicode[STIX]{x1D70C}_{S}$ to ${X\unicode[STIX]{x0030A}}$ . The assumption of topological normality guarantees that a general fiber of ${\unicode[STIX]{x1D70C}\unicode[STIX]{x0030A}}_{S}$ is connected.

The $\unicode[STIX]{x1D70C}$ -basic $C^{\infty }$ -differential forms in $j_{\ast }\mathscr{A}_{{X\unicode[STIX]{x0030A}}}^{\bullet }$ make up a differential (bigraded) subalgebra $\mathscr{A}_{X,\unicode[STIX]{x1D70C}}^{\bullet }$ that is a fine resolution of the constant sheaf $\mathbb{C}_{X}$ on $X$ (see Verona [Reference VeronaVer71] and [Reference Goresky and PardonGP02, Theorem 4.2]). It has the property that for all $S\in {\mathcal{S}}$ , $i_{S}^{-1}\mathscr{A}_{X,\unicode[STIX]{x1D70C}}^{\bullet }=\mathscr{A}_{S}^{\bullet }$ . Its holomorphic part defines a subcomplex $\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{\bullet }\subseteq j_{\ast }\unicode[STIX]{x1D6FA}_{{X\unicode[STIX]{x0030A}}}^{\bullet }$ with a similar property: $i_{S}^{-1}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{\bullet }=\unicode[STIX]{x1D6FA}_{S}^{\bullet }$ . This is also a resolution of the constant sheaf $\mathbb{C}_{X}$ and we can regard $\mathscr{A}_{X,\unicode[STIX]{x1D70C}}^{\bullet }$ as a double complex which resolves it. So, $(\mathscr{A}_{X,\unicode[STIX]{x1D70C}}^{p,\bullet },\bar{\unicode[STIX]{x2202}})$ resolves $\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{p}$ and we have a Hodge–De Rham spectral sequence

$$\begin{eqnarray}E_{2}^{p,q}=H^{q}(X,\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{p})\Rightarrow H^{p+q}(X,\mathbb{C}).\end{eqnarray}$$

If we are in the complex-projective setting, then one may wonder whether this spectral sequence degenerates and yields the Hodge filtration of the mixed Hodge structure on $H^{\bullet }(X)$ . This is not so in general (there exist examples for which $H^{0}(X,\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{p})=0\not =F^{p}H^{p}(X)$ ), but if it is at least true that the limit filtration of this spectral sequence refines the Hodge filtration, then we would have a generalization of Theorem 2.8 below and would probably also end up with a simpler proof of it.

Note that a morphism $\unicode[STIX]{x1D70B}\,:\,(\tilde{X},\tilde{{\mathcal{S}}},\tilde{\unicode[STIX]{x1D70C}})\rightarrow (X,{\mathcal{S}},\unicode[STIX]{x1D70C})$ determines a map of sheaf complexes $\unicode[STIX]{x1D70B}^{-1}\mathscr{A}_{X,\unicode[STIX]{x1D70C}}^{\bullet }\rightarrow \mathscr{A}_{\tilde{X},\tilde{\unicode[STIX]{x1D70C}}}^{\bullet }$ which induces the usual map $\unicode[STIX]{x1D70B}^{\ast }\,:\,H^{\bullet }(X;\mathbb{C})\rightarrow H^{\bullet }(\tilde{X};\mathbb{C})$ on cohomology.

We extend the notions introduced in §2.1 to this stratified germ type of setting.

Given a flat $\unicode[STIX]{x1D70C}$ -connection on ${\mathcal{F}}$ , then its $\unicode[STIX]{x1D70C}$ -flat local sections define a subsheaf of $j_{\ast }{\mathcal{F}}$ , but this subsheaf can be zero on certain strata and is probably of little interest unless the holonomies are trivial. More relevant to us is the subsheaf of ${\mathcal{O}}_{X,\unicode[STIX]{x1D70C}}$ -algebras ${\mathcal{E}}\!nd({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})\subseteq j_{\ast }{\mathcal{E}}\!nd({\mathcal{F}})$ of $\unicode[STIX]{x1D70C}$ -flat local endomorphisms of $j_{\ast }{\mathcal{F}}$ , at least when the flat $\unicode[STIX]{x1D70C}$ -connection on ${\mathcal{F}}$ is isoholonomic, for then $i_{S}^{-1}{\mathcal{E}}\!nd({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})$ is locally like ${\mathcal{E}}\!nd_{{\mathcal{O}}_{S}}({\mathcal{O}}_{S}^{r})$ (it is a sheaf of Azumaya ${\mathcal{O}}_{S}$ -algebras).

Note that if $\unicode[STIX]{x1D70B}\,:\,(\tilde{X},\tilde{{\mathcal{S}}},\tilde{\unicode[STIX]{x1D70C}})\rightarrow (X,{\mathcal{S}},\unicode[STIX]{x1D70C})$ is a morphism of rigidified stratified, topological normal varieties, then a flat $\unicode[STIX]{x1D70C}$ -connection on ${\mathcal{F}}$ determines one on the pull-back of ${\mathcal{F}}$ along ${\unicode[STIX]{x1D70B}\unicode[STIX]{x0030A}}\,:\,\{\rightarrow {X\unicode[STIX]{x0030A}}$ (that we simply denote by $\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ ): if $\unicode[STIX]{x1D70B}$ maps $\tilde{S}\in \tilde{{\mathcal{S}}}$ to $S\in {\mathcal{S}}$ , then $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}_{S}}$ determines in an obvious manner a flat connection along the fibers of $\{_{\tilde{S}}$ and the resulting system has the required properties. It is isoholonomic when $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}$ is.

In the situation of the last clause of Proposition 2.5, we find that $\operatorname{c}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})$ lifts to an integral class. But we will see that this is not so in general.

Remark 2.6. We shall later want to work with Chern characters $ch_{k}$ rather than with Chern classes $c_{k}$ . Since we always use $\mathbb{Q}$ -vector spaces as coefficients, there is no loss of information here: $ch_{k}$ is a universal polynomial of weighted degree $k$ with rational coefficients in $c_{1},\ldots ,c_{k}$ and vice versa. These Chern characters can also be obtained via an Atiyah class, which is perhaps closer in the spirit of algebraic geometry, albeit that they then come to be realized as De Rham classes. An isoholonomic flat $\unicode[STIX]{x1D70C}$ -connection defines a natural lift of the Atiyah class of ${\mathcal{F}}$ , $\operatorname{At}({\mathcal{F}})\in H^{1}({X\unicode[STIX]{x0030A}},\unicode[STIX]{x1D6FA}_{{X\unicode[STIX]{x0030A}}}^{1}\otimes {\mathcal{E}}\!nd({\mathcal{F}}))$ , to an element $\operatorname{At}({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})\in H^{1}(X,\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{1}\otimes {\mathcal{E}}\!nd({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}}))$ . A representative as a 1-Čech cocycle is obtained from the collection $(U_{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}}$ in the proof above: it is given by $U_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=U_{\unicode[STIX]{x1D6FC}}\cap U_{\unicode[STIX]{x1D6FD}}\mapsto \unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D6FC}}$ . We then define the twisted Goresky–Pardon Chern character as the image of $\operatorname{At}({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})$ under the map

$$\begin{eqnarray}\displaystyle H^{1}(X,\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{1}\otimes {\mathcal{E}}\!nd({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})) & \rightarrow & \displaystyle \bigoplus _{k=0}^{\infty }H^{k}(X,\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D70C}}^{k}),\nonumber\\ \displaystyle A & \mapsto & \displaystyle \operatorname{Tr}(\exp (-A))=\mathop{\sum }_{k=0}^{\infty }\frac{\operatorname{Tr}((-A)^{\cup k})}{k!}.\nonumber\end{eqnarray}$$

This class is closed for all the differentials in the Hodge–De Rham spectral sequence and then yields $(2\unicode[STIX]{x1D70B}\sqrt{-1})^{k}\operatorname{ch}_{k}({\mathcal{F}},\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D70C}})$ . This observation leads us to the following result.

The notion of a resolution of a stratified variety that appears in the formulation of the theorem above expresses the fact that such a variety is equisingular along strata in a rather strong sense. Among other things, it can be shown to imply Whitney’s $(a)$ condition. We define this notion and prove the theorem in the next subsection.

Stratified resolutions

We begin with noting that if on a complex manifold $Y$ there is given a normal crossing divisor $D$ , then $Y$ acquires a natural stratification, where a stratum is a connected component of the locus where for some integer $l\geqslant 0$ exactly $l$ local branches of $D$ meet. With $D$ given, we will often write $Y^{(l)}$ for the normalization of the locus where at least $l$ branches of $D$ meet (so that $Y^{(0)}=Y$ ). This is clearly a complex manifold. If $E$ is a connected component of $Y^{(l)}$ , then the locus where ${>}l$ branches of $D$ meet traces out on $E$ a normal crossing divisor, which is simple when $D$ is and whose normalization is contained in $Y^{(l+1)}$ . When $l>0$ , then, for the same reason, $E$ naturally maps to a number of connected components of $Y^{(l-1)}$ . When $D$ is simple, this number is $l$ and the maps are embeddings.

Let $(X,{\mathcal{S}})$ be a stratified analytic variety and assume that the normalization of $X$ is a homeomorphism.

Note that then any stratum $S\in {\mathcal{S}}$ inherits such a structure in the sense that the collection $\{\unicode[STIX]{x1D70B}_{S^{\prime }}\}_{S^{\prime }\leqslant S}$ defines a ${\mathcal{S}}|\overline{S}$ -resolution of $\overline{S}$ .

In order to prove Theorem 2.8, we first show how the above notion gives rise to a simplicial resolution that can be used to compute the cohomology of $X$ and its mixed Hodge structure, when that makes sense.

Obviously, the collection of $\unicode[STIX]{x1D70B}_{S}\,:\,\tilde{S}\rightarrow \overline{S}$ , where $S\in {\mathcal{S}}$ runs over the open strata, defines a resolution $\unicode[STIX]{x1D70B}\,:\,\tilde{X}\rightarrow X$ of $X$ whose exceptional set is a normal crossing divisor. So, $X$ can be regarded as a quotient space of $\tilde{X}$ with the identifications taking place over the strata of depth ${\geqslant}1$ . Let $S>S^{\prime }$ be a pair of incident strata whose depths differ by 1. When we regard $\overline{S}$ as a quotient of $\tilde{S}$ , then the identification over $S^{\prime }$ is exhibited by $\unicode[STIX]{x1D70B}_{S^{\prime }}^{S}\,:\,\tilde{S}[S^{\prime }]\rightarrow \tilde{S}^{\prime }$ . In order to let all such identifications take place by means of morphisms between smooth varieties, it is best to replace $\tilde{S}[S^{\prime }]$ by its normalization. This means that we should do this for every connected component of this normalization. It is then wise to remember that these connected components are glued to each other in $\tilde{S}[S^{\prime }]$ . We may continue this process with any stratum of depth 2 and finally end up with a small category $\mathscr{S}$ of compact complex manifolds over $X$ that has $X$ as a direct limit in the category of topological spaces. Here is a more precise description of $\mathscr{S}$ .

An object of $\mathscr{S}$ is a connected component $E$ of $\tilde{S}^{(l)}$ for some $S\in {\mathcal{S}}$ and some $l\geqslant 0$ . So, when we regard $E$ as a subvariety of $\tilde{S}$ , it is the closure of a stratum. We describe two types of basic morphisms $E\rightarrow E^{\prime }$ between two such objects and stipulate that these generate the $\mathscr{S}$ -morphisms. The first one is when $E^{\prime }$ is obtained from $E$ by forgetting one of the $l$ irreducible components of $D_{\tilde{S}}$ which contains $E$ and $E\rightarrow E^{\prime }$ is the obvious embedding. The other is defined only when $E$ is contained in $\tilde{S}[S^{\prime }]$ for some $S^{\prime }<S$ . Then $E^{\prime }:=\unicode[STIX]{x1D70B}_{S}^{S^{\prime }}(E)$ is a connected component of $\tilde{S}^{\prime (l^{\prime })}$ for some $l^{\prime }$ and the resulting map $E\rightarrow E^{\prime }$ is the other type of basic morphism.

Now recall that the nerve of the small category $\mathscr{S}$ is a simplicial set whose $n$ -simplices are chains of length $n\,:\,E_{\bullet }=(E_{0}\rightarrow E_{1}\rightarrow \cdots \rightarrow E_{n})$ in $\mathscr{S}$ . We then obtain a simplicial space $X_{\bullet }$ by taking for $X_{n}$ the disjoint union of the objects $\operatorname{in}(E_{\bullet })$ , where $E_{\bullet }$ runs over the chains of length $n$ in $\mathscr{S}$ and $\operatorname{in}(E_{\bullet })$ stands for the first term of such a chain. (So, the connected components of $X_{n}$ are objects of $\mathscr{S}$ , but the indexing is by the $n$ -simplices, so that several copies of the same $\mathscr{S}$ -object may appear.) Its geometric realization $|X_{\bullet }|$ (which is obtained in a standard fashion as a quotient of the disjoint union of the products $|\unicode[STIX]{x1D6E5}^{l}|\times X_{l}$ ) is a space over $X$ and this structural map is a homotopy equivalence. It can be used for cohomological descent: the face maps $\unicode[STIX]{x2202}_{i}\,:\,X_{n}\rightarrow X_{n-1}$ , $0\leqslant i\leqslant n$ , are used to define a double cochain complex

$$\begin{eqnarray}C^{\bullet }(X_{\bullet })\,:\,0\rightarrow C^{\bullet }(X_{0})\rightarrow C^{\bullet }(X_{1})\rightarrow \cdots\end{eqnarray}$$

and the obvious chain homomorphism from $C^{\bullet }(X)$ to the associated simple complex $sC^{\bullet }(X_{\bullet })$ induces an isomorphism on (integral) cohomology. In particular, we have spectral sequence

(2.1) $$\begin{eqnarray}E_{1}^{r,s}=H^{s}(X_{r})\Rightarrow H^{r+s}(X).\end{eqnarray}$$

We note that the edge homomorphism $H^{\bullet }(X)\cong H^{\bullet }(sC^{\bullet }(X_{\bullet }))\rightarrow H^{\bullet }(X_{0})=\bigoplus _{E\in \mathscr{S}}H^{\bullet }(E)$ is induced by the obvious map $\bigsqcup _{E\in \mathscr{S}}E=X_{0}\rightarrow X$ .

Suppose that we are in the algebraic setting, so that varieties and morphisms are complex-algebraic. Then $H^{\bullet }(X)$ carries a mixed Hodge structure and we can use this construction to identify that structure: the above spectral sequence is one of mixed Hodge structures. This implies that when $X$ is compact, it degenerates at $E_{2}$ (all higher differentials are zero since their source and target when nonzero have different weights) and yields the weight filtration:

(2.2) $$\begin{eqnarray}\operatorname{gr}_{s}^{W}H^{r+s}(X;\mathbb{Q})=E_{2}^{r,s}=H(H^{s}(X_{r-1};\mathbb{Q})\rightarrow H^{s}(X_{r};\mathbb{Q})\rightarrow H^{s}(X_{r+1};\mathbb{Q})).\end{eqnarray}$$

Moreover, if we use $A^{\bullet }(M)$ to denote the $\mathbb{C}$ -valued De Rham complex of a complex-algebraic manifold $M$ and $F^{\bullet }A^{\bullet }(M)$ its Hodge filtration, then the Hodge filtration of $sA^{\bullet }(X_{\bullet })$ defines the Hodge filtration of $H^{\bullet }(X;\mathbb{C})$ .

Perhaps the simplest nontrivial example is when $X$ has only two strata: $S$ and $X-S$ and $\unicode[STIX]{x1D70B}\,:\,\tilde{X}\rightarrow X$ is a resolution with $\unicode[STIX]{x1D70B}^{-1}S$ nonsingular. Then the complex $C^{\bullet }(X_{\bullet })$ is just $0\rightarrow C^{\bullet }(\tilde{X})\oplus C^{\bullet }(S)\rightarrow C^{\bullet }(\unicode[STIX]{x1D70B}^{-1}S)\rightarrow 0$ and the associated exact sequence

$$\begin{eqnarray}\cdots \rightarrow H^{s-1}(\unicode[STIX]{x1D70B}^{-1}S)\rightarrow H^{s}(X)\rightarrow H^{s}(\tilde{X})\oplus H^{s}(\unicode[STIX]{x1D70B}^{-1}S)\rightarrow H^{s}(\unicode[STIX]{x1D70B}^{-1}S)\rightarrow \cdots\end{eqnarray}$$

yields the weight filtration: $W_{s-2}H^{s}(X)=0$ , $W_{s-1}H^{s}(X)$ is the image of the map $H^{s-1}(\unicode[STIX]{x1D70B}^{-1}S;\mathbb{Q})\rightarrow H^{s}(X;\mathbb{Q})$ and $W_{s}H^{s}(X)=H^{s}(X;\mathbb{Q})$ .

Review of the Baily–Borel compactification

Let ${\mathcal{G}}$ be a connected reductive complex algebraic group that is defined over $\mathbb{R}$ . Write $G$ (respectively, $G_{\mathbb{C}}$ ) for ${\mathcal{G}}(\mathbb{R})$ (respectively, ${\mathcal{G}}(\mathbb{C})$ ) endowed with the Hausdorff topology. We assume that $G$ has compact center and that the symmetric space $\mathbb{X}$ of $G$ is endowed with a $G$ -invariant complex structure. To say that $\mathbb{X}$ is the symmetric space $\mathbb{X}$ of $G$ means that for every $x\in \mathbb{X}$ the stabilizer $G_{x}$ is a maximal compact subgroup of $G$ and to say that $\mathbb{X}$ comes with a $G$ -invariant complex structure amounts to the property that $G_{x}$ contains an embedded copy of the circle group $\operatorname{U}(1)$ in its center whose action on $T_{x}\mathbb{X}$ defines its complex structure (it is a nontrivial action by scalars of unit norm). This makes $\mathbb{X}$ a bounded symmetric domain. It appears naturally as an open $G$ -orbit in a complex projective manifold $\check{\mathbb{X}}$ , called the compact dual of $\mathbb{X}$ , on which $G_{\mathbb{C}}$ acts transitively. It has the property that for $x\in \mathbb{X}$ , the $G_{\mathbb{C}}$ -stabilizer $G_{\mathbb{C},x}$ is simply the complexification of $G_{x}$ .

We now assume that ${\mathcal{G}}$ is defined over $\mathbb{Q}$ and let $\unicode[STIX]{x1D6E4}\subset G(\mathbb{Q})$ be an arithmetic subgroup. For what follows, the passage to a subgroup of $\unicode[STIX]{x1D6E4}$ of finite index will be harmless, and so we will assume from the outset that $\unicode[STIX]{x1D6E4}$ is neat. This means that for every finite-dimensional representation $\unicode[STIX]{x1D70C}\,:\,G_{\mathbb{C}}\rightarrow \operatorname{GL}(n,\mathbb{C})$ , the subgroup of $\mathbb{C}^{\times }$ generated by the eigenvalues of elements of $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E4})$ has no torsion (actually it suffices to verify this for just one faithful representation). This implies that the arithmetically defined subquotients of $\unicode[STIX]{x1D6E4}$ are torsion-free. The action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{X}$ is then proper and free, so that the orbit space $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ is a complex manifold. The Baily–Borel compactification, which we will presently recall, shows that $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ has even the structure of a nonsingular quasi-projective variety.

A central role in the Baily–Borel theory is played by the collection $\mathscr{P}_{\max }=\mathscr{P}_{\max }(G)$ of maximal proper parabolic subgroups of ${\mathcal{G}}$ defined over $\mathbb{Q}$ , and so let us fix some $P\in \mathscr{P}_{\max }$ . We review the structure of $P$ and the way it acts on $\mathbb{X}$ . Its unipotent radical $R_{u}(P)\subseteq P$ is at most two-step unipotent: if $U_{P}\subseteq R_{u}(P)$ denotes its center (a nontrivial vector group), then $V_{P}:=R_{u}(P)/U_{P}$ is also a (possibly trivial) vector group. Adopting the convention to denote the associated Lie algebras by the corresponding Fraktur font, then the Lie bracket defines an antisymmetric bilinear map $\mathfrak{v}_{P}\times \mathfrak{v}_{P}\rightarrow \mathfrak{u}_{P}$ . This map is equivariant with respect to the adjoint action of $P$ on these vector spaces. Note that $P$ acts on $\mathfrak{u}_{P}$ and $\mathfrak{v}_{P}$ through its Levi quotient $L_{P}:=P/R_{u}(P)$ . The reductive group $L_{P}$ has in $\mathfrak{u}_{P}$ a distinguished open orbit that is a strictly convex cone $C_{P}$ having the property that if we exponentiate $\sqrt{-1}C_{P}$ to a semigroup in $G_{\mathbb{C}}$ , then this semigroup leaves $\mathbb{X}$ invariant (think of the upper half plane in $\mathbb{C}$ that is invariant under the semigroup of translations in $\sqrt{-1}\mathbb{R}_{{>}0}$ ). The $G$ -stabilizer of $\mathfrak{u}_{P}$ is $P$ and so $\mathfrak{u}_{P}$ determines $P$ . This gives rise to a partial order ${\leqslant}$ on $\mathscr{P}_{\max }$ by stipulating that $Q\leqslant P$ in case $\mathfrak{u}_{Q}\subseteq \mathfrak{u}_{P}$ (in $\mathfrak{g}$ ). This last property is equivalent to $C_{Q}\subseteq \overline{C}_{P}$ . The other (nonmaximal) $\mathbb{Q}$ -parabolic subgroups of $G$ are obtained from chains in $\mathscr{P}_{\max }$ : for a chain $P_{0}<P_{1}<\cdots <P_{n}$ in $\mathscr{P}_{\max }$ , $P:=P_{0}\cap \cdots \cap P_{n}$ is a $\mathbb{Q}$ -parabolic subgroup. It has the property that its unipotent radical contains the unipotent radical of each $P_{i}\,:\,\bigcup _{i=1}^{n}R_{u}(P_{i})\subseteq R_{u}(P)$ .

For $P\in \mathscr{P}_{\max }$ , the $\mathbb{Q}$ -split center $A_{P}$ of $L_{P}$ is isomorphic as such to the multiplicative group (and so $A_{P}\cong \mathbb{R}^{\times }$ ). It acts on $\mathfrak{v}_{P}$ by a faithful character (multiplication by scalars) and on $\mathfrak{u}_{P}$ by the square of that character (so that it indeed preserves $C_{P}$ ). The horizontal subgroup $M_{P}^{\mathit{h}}\subseteq L_{P}$ (for some authors, the superscript stands for hermitian) is the kernel of the action of $L_{P}$ on $\mathfrak{u}_{P}$ . This is a reductive subgroup defined over $\mathbb{Q}$ with compact center. The centralizer of $M_{P}^{\mathit{h}}$ in $L_{P}$ is a reductive $\mathbb{Q}$ -subgroup whose commutator subgroup we denote by $M_{P}^{\ell }$ (we like to think that the symbol $\ell$ should refer to link rather than linear; the explanation for this terminology will become clear below). This group acts in such a manner on the real projectivization of $C_{P}$ (in the projective space of $\mathfrak{u}_{P}$ ) that the latter is the symmetric space of $M_{P}^{\ell }$ . So, we may regard $C_{P}$ as the symmetric space of $L_{P}^{\ell }:=M_{P}^{\ell }.A_{P}$ . The group $L_{P}^{\ell }$ supplements $M_{P}^{\mathit{h}}$ in $L_{P}$ up to a finite central subgroup. We denote the preimage of $L_{P}^{\ell }$ in $P$ by $P^{\ell }$ .

The action of $P$ on $\mathbb{X}$ is still transitive. Important for what follows is that the formation of the $P^{\ell }$ -orbit space of $\mathbb{X}$ remains in the holomorphic category: it defines a holomorphic submersion of complex manifolds $\mathbb{X}\rightarrow \mathbb{X}(P)$ , with $\mathbb{X}(P)$ appearing as the symmetric domain of $M_{P}^{\mathit{h}}$ . This is called a rational boundary component of $\mathbb{X}$ . The $P^{\ell }$ -orbits in $\mathbb{X}$ (so the fibers of $\mathbb{X}\rightarrow \mathbb{X}(P)$ ) are also orbits of the semisubgroup $R_{u}(P)+\exp (\sqrt{-1}C_{P})\subseteq G_{\mathbb{C}}$ in $\check{\mathbb{X}}$ and this description is essentially an abstract way of realizing $\mathbb{X}$ as a Siegel domain of the third kind. To be precise, we have a natural factorization of $\unicode[STIX]{x1D70C}_{P}\,:\,\mathbb{X}\rightarrow \mathbb{X}(P)$ :

(3.1)

where the first map $\unicode[STIX]{x1D70C}_{P}^{\prime }$ is a bundle of tube domains (a ‘torsor’ over $\mathbb{X}(P)^{\prime }$ for the semigroup $\exp (\mathfrak{u}+(\sqrt{-1}C_{P}))$ ) and the second map $\unicode[STIX]{x1D70C}_{P}^{\prime \prime }$ is a principal bundle of the vector group $V_{P}=\exp (\mathfrak{v}_{P})$ (so a bundle of affine spaces). The latter has also the structure of a complex affine space bundle, but beware that this complex structure on a fiber (which can be given as a complex structure on its translation space $\mathfrak{v}$ ) will in general vary with the base point. The map $M_{P}^{\mathit{h}}\rightarrow L_{P}/L_{P}^{\ell }=P/P^{\ell }$ is an isogeny: it is onto and has finite kernel. We write $G_{P}$ for $P/P^{\ell }$ . The action of $M_{P}^{\mathit{h}}$ on $\mathbb{X}(P)$ is through this quotient and we prefer to regard $\mathbb{X}(P)$ as the symmetric space of the quotient $G_{P}$ of $P$ rather than of the subquotient $M_{P}^{\mathit{h}}$ of $P$ (see the example of the symplectic group below).

Every $Q\in \mathscr{P}_{\max }(G)$ with $Q>P$ has by definition the property that $\mathfrak{u}_{Q}\supset \mathfrak{u}_{P}$ . But it is then even true that $Q^{\ell }\supset P^{\ell }$ and so the projection $\unicode[STIX]{x1D70C}_{Q}\,:\,\mathbb{X}\rightarrow \mathbb{X}(Q)$ factors through $\unicode[STIX]{x1D70C}_{P}\,:\,\mathbb{X}\rightarrow \mathbb{X}(P)$ via a morphism that we shall denote by $\unicode[STIX]{x1D70C}_{Q_{/P}}\,:\,\mathbb{X}(P)\rightarrow \mathbb{X}(Q)$ . The latter can be understood as the formation of a rational boundary component of $\mathbb{X}(P)$ . Indeed, $Q$ defines a maximal proper parabolic subgroup of $G_{P}$ , namely the image of $Q\cap L_{P}$ in $L_{P}/L_{P}^{\ell }=G_{P}$ (that we shall denote by $Q_{/P}$ ). This identifies $\mathscr{P}_{\max }(G_{P})$ as a partially ordered set with $\mathscr{P}_{\max }(G)_{{>}P}$ . The unipotent radical of $Q_{/P}$ is the image of $R_{u}(P)\cap R_{u}(Q)$ in $Q_{/P}$ . Its center $U_{Q_{/P}}$ is the image of $U_{Q}$ , $U_{Q}/U_{Q}\cap R_{u}(P)$ . Similarly, the cone $C_{Q_{/P}}\subseteq \mathfrak{u}_{Q_{/P}}$ is the image of $C_{Q}\subseteq \mathfrak{u}_{Q}$ under the projection $\mathfrak{u}_{Q}\rightarrow \mathfrak{u}_{Q}/\mathfrak{u}_{Q}\cap R_{u}(\mathfrak{p})\cong \mathfrak{u}_{Q_{/P}}$ :

We define the Satake extension of $\mathbb{X}$ as a ringed space. As a set, it is the disjoint union

$$\begin{eqnarray}\mathbb{X}^{\text{bb}}:=\mathbb{X}\sqcup \bigsqcup _{P\in \mathscr{P}_{\max }}\mathbb{X}(P).\end{eqnarray}$$

It is endowed with the horocyclic topology: the topology generated by the open subsets of $\mathbb{X}$ and the subsets $\unicode[STIX]{x1D6FA}^{\text{bb}_{P}}$ , where $P\in \mathscr{P}_{\max }$ and $\unicode[STIX]{x1D6FA}\subseteq \mathbb{X}$ is open and invariant under both the semigroup $\sqrt{-1}C_{P}$ and the group $\unicode[STIX]{x1D6E4}\cap P^{\ell }$ , and

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}^{\text{bb}_{P}}:=\unicode[STIX]{x1D6FA}\sqcup \bigsqcup _{Q\in \mathscr{P}_{\max };Q\leqslant P}\unicode[STIX]{x1D70C}_{Q}(\unicode[STIX]{x1D6FA}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D6E4}\cap R_{u}(P)$ is cocompact in $R_{u}(P)$ , we may replace here invariance under $\unicode[STIX]{x1D6E4}\cap P^{\ell }$ by invariance under $R_{u}(P).(\unicode[STIX]{x1D6E4}\cap P^{\ell })$ (but not in general by invariance under $P^{\ell }$ ). Yet this topology is independent of $\unicode[STIX]{x1D6E4}$ : it only depends on the $\mathbb{Q}$ -structure on ${\mathcal{G}}$ . This construction is natural in the sense that the closure of any rational boundary component in $\mathbb{X}^{\text{bb}}$ can be identified with its Satake extension. The structure sheaf ${\mathcal{O}}_{\mathbb{ X}^{\text{bb}}}$ is the sheaf of complex-valued continuous functions that are holomorphic on every stratum $\mathbb{X}(P)$ . It is clear that $\unicode[STIX]{x1D6E4}$ acts on this ringed space. The main theorem of Baily–Borel asserts among other things that the orbit space $(\mathbb{X}_{\unicode[STIX]{x1D6E4}}^{\text{bb}},{\mathcal{O}}_{\mathbb{ X}_{\unicode[STIX]{x1D6E4}}^{\text{bb}}})$ is as a ringed space a normal compact analytic space that underlies the structure of a normal projective variety and, by a theorem of Chow, this projective structure is then unique. Moreover, the decomposition of $\mathbb{X}^{\text{bb}}$ into $\mathbb{X}$ and its rational boundary components defines a decomposition of $\mathbb{X}_{\unicode[STIX]{x1D6E4}}^{\text{bb}}$ into nonsingular subvarieties (strata) such that the closure of any of these is a union thereof. Any stratum is of the same type as $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ : it is of the form $\mathbb{X}(P)_{\unicode[STIX]{x1D6E4}(G_{P})}$ and hence has its own Baily–Borel compactification. The preceding shows that the Baily–Borel compactification of a stratum maps homeomorphically onto its closure in $\mathbb{X}_{\unicode[STIX]{x1D6E4}}^{\text{bb}}$ . This map is also a morphism of varieties (that could be an isomorphism, but it is conceivable that this closure is not normal). This shows among other things the following result.

Satake extension of automorphic bundles

Let ${\mathcal{F}}$ be an automorphic vector bundle on $\mathbb{X}$ , that is, a complex vector bundle on $\mathbb{X}$ endowed with a $G$ -action lifting the one on $\mathbb{X}$ in such a manner that for some (and hence for any) $x\in \mathbb{X}$ the copy of $\operatorname{U}(1)$ in the stabilizer $G_{x}$ acts also complex linearly on the fiber ${\mathcal{F}}(x)$ . Such a vector bundle is completely given by the action of $G_{x}$ on the complex vector space ${\mathcal{F}}(x)$ and, conversely, any finite-dimensional complex representation of $G_{x}$ defines such a vector bundle. The bundle ${\mathcal{F}}$ with its $G$ -action extends to the compact dual $\check{\mathbb{X}}$ as a vector bundle with $G_{\mathbb{C}}$ -action and this extension (which we denote by $\check{{\mathcal{F}}}$ ) is unique. This is because the $G_{x}$ -action on the complex vector space ${\mathcal{F}}(x)$ extends to one of the complexifications $G_{x,\mathbb{C}}$ of $G_{x}$ , and $G_{x,\mathbb{C}}$ is just the $G_{\mathbb{C}}$ -stabilizer of $x$ . Since the $G_{\mathbb{C}}$ -bundle $\check{{\mathcal{F}}}$ is defined in the holomorphic category, it follows that ${\mathcal{F}}$ comes with a $G$ -invariant holomorphic structure.

Given $x\in \mathbb{X}$ , the compactness of $G_{x}$ implies that $G_{x}$ leaves invariant an inner product in the fiber ${\mathcal{F}}(x)$ . This inner product then extends in a unique manner to a $G$ -invariant inner product $h$ on ${\mathcal{F}}$ . As is well known, we then have a unique hermitian connection $\unicode[STIX]{x1D6FB}$ on $({\mathcal{F}},h)$ whose $(0,1)$ -part is zero on local holomorphic sections. This connection is of course also $G$ -invariant. It is in fact independent of $h$ . This is clear when ${\mathcal{F}}(x)$ is irreducible as a representation of $G_{x}$ , for then the inner product is unique up to a scalar and the general case then follows from this by decomposing ${\mathcal{F}}(x)$ into irreducible subrepresentations. So, we have canonically associated Chern forms $\operatorname{C}_{n}({\mathcal{F}})=\operatorname{C}_{n}({\mathcal{F}},\unicode[STIX]{x1D6FB})$ on $\mathbb{X}$ . Such a form is harmonic relative to a $G$ -invariant metric on $\mathbb{X}$ , is $G$ -invariant and of Hodge bidegree $(n,n)$ . The $G$ -equivariance allows us to descend all of this to $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ , so that we get a holomorphic bundle ${\mathcal{F}}_{\unicode[STIX]{x1D6E4}}$ with connection on $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ whose Chern forms pull back to the ones of $({\mathcal{F}},\unicode[STIX]{x1D6FB})$ . The $G$ -invariant connection $\unicode[STIX]{x1D6FB}$ will in general not extend to $\check{{\mathcal{F}}}$ and neither will the associated Chern forms.

The quotient of ${\mathcal{F}}$ by the $\unicode[STIX]{x1D6E4}$ -action gives a holomorphic vector bundle ${\mathcal{F}}_{\unicode[STIX]{x1D6E4}}$ on $\mathbb{X}_{\unicode[STIX]{x1D6E4}}$ . The following corollary somewhat sharpens the main result of [Reference Goresky and PardonGP02].

Example 3.5. A symplectic group and its Hodge bundle. Let a finite-dimensional real vector space $V$ endowed with a nondegenerate symplectic form $a\,:\,V\times V\rightarrow \mathbb{R}$ be given. The automorphism group of $(V,a)$ defines an almost simple algebraic group defined over $\mathbb{R}$ whose group of real (respectively, complex) points (endowed with the Hausdorff topology) is $\operatorname{Sp}(V)$ (respectively, $\operatorname{Sp}(V_{\mathbb{C}})$ ). Let $h\,:\,V_{\mathbb{C}}\times V_{\mathbb{C}}\rightarrow \mathbb{C}$ be the hermitian form defined by $h(v,v^{\prime }):=\sqrt{-1}a_{\mathbb{C}}(v,\overline{v}^{\prime })$ . It has signature $(g,g)$ . The associated symmetric space of $\operatorname{Sp}(V)$ and its compact dual are obtained as follows: $\check{\mathbb{X}}=\check{\mathbb{H}}(V)$ is the locus in the Grassmannian $\operatorname{Gr}_{g}(V_{\mathbb{C}})$ which parameterizes the $g$ -dimensional subspaces $F\subseteq V_{\mathbb{C}}$ that are totally isotropic relative to $a_{\mathbb{C}}$ and $\mathbb{X}=\mathbb{H}(V)$ is the open subset of $\check{\mathbb{H}}(V)$ parameterizing those $F$ on which in addition $h$ is positive definite. The group $\operatorname{Sp}(V)$ indeed acts transitively on $\mathbb{H}(V)$ and the stabilizer of any $[F]\in \mathbb{H}(V)$ restricts isomorphically to the unitary group $U(F)$ , which is a maximal compact subgroup of $\operatorname{Sp}(V)$ . The restriction of the tautological rank- $g$ bundle on $\operatorname{Gr}_{g}(V_{\mathbb{C}})$ to $\mathbb{H}(V)$ (respectively, $\check{\mathbb{H}}(V)$ ) is an automorphic bundle ${\mathcal{F}}={\mathcal{F}}_{V}$ (respectively its natural extension $\check{{\mathcal{F}}}$ to $\check{\mathbb{H}}(V)$ ). For $[F]\in \mathbb{H}(V)$ , we have $V_{\mathbb{C}}=F\oplus \overline{F}$ and so $F$ is the $(1,0)$ -part of a Hodge structure on $V$ of weight 1 polarized by $a$ . Thus, $\mathbb{H}(V)$ also parameterizes polarized Hodge structures on $V$ of this type. For this reason, ${\mathcal{F}}$ is often called the Hodge bundle. Notice that $h$ defines on ${\mathcal{F}}$ an inner product (that we continue to denote by $h$ ).

Now assume that $V$ and $a$ are defined over $\mathbb{Q}$ , so that our group is also defined over $\mathbb{Q}$ . A maximal proper $\mathbb{Q}$ -parabolic subgroup $P\subseteq \operatorname{Sp}(V)$ is the $\operatorname{Sp}(V)$ -stabilizer of a nonzero isotropic subspace $I\subseteq V$ defined over $\mathbb{Q}$ and vice versa. So, $\mathscr{P}_{\max }$ may be identified with the set $\mathscr{I}(V)$ of nonzero $\mathbb{Q}$ -isotropic subspaces of $V$ . We will therefore index our objects accordingly.

Let $I\in \mathscr{I}(V)$ and write $V_{I}^{\prime }$ for $V/I$ and $V_{I}\subseteq V_{I}^{\prime }$ for $I^{\bot }/I$ . Note that the symplectic form identifies $V_{I}^{\prime }$ with the dual of $I^{\bot }$ and induces on $V_{I}$ a nondegenerate symplectic form. Then the unipotent radical $R_{u}(P_{I})$ of $P_{I}$ is the subgroup that acts trivially on $I$ and $V_{I}$ ; note that it then also acts trivially on $V/I^{\bot }$ . This identifies the Levi quotient $L_{I}$ of $P_{I}$ with $\operatorname{GL}(I)\times \operatorname{Sp}(V_{I})$ . The center $U_{I}$ of $R_{u}(P_{I})$ is the subgroup that acts trivially on $I^{\bot }$ (or, equivalently, on $V_{I}^{\prime }$ ). Its (abelian) Lie algebra $\mathfrak{u}_{I}$ can be identified with $\operatorname{Sym}^{2}I\subseteq \operatorname{Sym}^{2}V\cong \mathfrak{g}$ and $C_{I}\subseteq \operatorname{Sym}^{2}I$ is the cone of positive-definite elements. This identifies $R_{u}(P_{I})/U_{I}$ with a group of elements in $\operatorname{GL}(I^{\bot })$ that act trivially on both $I$ and $V_{I}$ ; this group is abelian with Lie algebra $\operatorname{Hom}(V_{I},I)$ (which we shall identify with $I\otimes V_{I}$ by means of the nondegenerate symplectic form on $V_{I}$ ). The central subgroup $A_{I}\subseteq P_{I}$ appears here as the group of scalars in $\operatorname{GL}(I)$ and hence is a copy of $\mathbb{R}^{\times }$ . The adjoint action of $L_{I}=\operatorname{GL}(I)\times \operatorname{Sp}(V_{I})$ on this Lie algebra is the obvious one. In terms of these isomorphisms, $M_{I}^{\mathit{h}}=\{\pm 1_{I}\}\times \operatorname{Sp}(V_{I})$ , $M_{I}^{\ell }=\operatorname{SL}(I)$ , $L_{I}^{\ell }=\operatorname{GL}(I)$ , $G_{I}=\operatorname{Sp}(V_{I})$ and $P_{I}^{\ell }\subseteq G_{I}$ is the group that acts as $\pm 1$ on $V_{I}$ .

We next describe the maps $\mathbb{X}\rightarrow \mathbb{X}(P_{I})^{\prime }\rightarrow \mathbb{X}(P_{I})$ . For this, we note that for $[F]\in \mathbb{H}(V)$ , the projection $F\rightarrow V_{I,\mathbb{C}}^{\prime }$ is into and the projection $F\rightarrow V_{\mathbb{C}}/I_{\mathbb{C}}^{\bot }\cong I_{\mathbb{C}}^{\ast }$ is onto with kernel $F\cap I_{\mathbb{C}}^{\bot }$ projecting isomorphically onto a subspace of $V_{I,\mathbb{C}}$ that defines an element of $\mathbb{H}(V_{I})$ . If we denote by $\mathbb{H}(V_{I}^{\prime })$ the subspace of Grassmannians of $V_{I,\mathbb{C}}^{\prime }$ parameterizing the subspaces whose intersection with $V_{I,\mathbb{C}}$ defines an element of $\mathbb{H}(V_{I})$ , then we obtain a diagram

where the maps are the obvious ones. It is clear that ${\mathcal{F}}_{V}$ is the $\unicode[STIX]{x1D70C}_{I}^{\prime }$ -pull-back of the tautological bundle ${\mathcal{F}}_{V_{I}^{\prime }}$ over $\mathbb{H}(V_{I}^{\prime })$ . We note that $\mathbb{H}(V_{I}^{\prime })\rightarrow \mathbb{H}(V_{I})$ is a principal bundle for the vector group $\operatorname{Hom}(V/I^{\bot },V_{I})\cong V_{I}\otimes I$ and that this $V_{I}\otimes I$ -action lifts to ${\mathcal{F}}_{V_{I}^{\prime }}$ .

Let us determine the flat $\unicode[STIX]{x1D70C}_{I}^{\prime \prime }$ -connection on ${\mathcal{F}}_{V_{I}^{\prime }}$ . First observe that any subspace $F^{\prime \prime }\subseteq V_{I}$ that defines an element of $\mathbb{H}(V_{I})$ determines a complex structure on $V_{I}$ characterized by the property that the $\mathbb{R}$ -linear isomorphism $V_{I}\subseteq V_{I,\mathbb{C}}\rightarrow V_{I,\mathbb{C}}/F^{\prime \prime }\cong \operatorname{Hom}_{\mathbb{C}}(F^{\prime \prime },\mathbb{C})$ is in fact $\mathbb{C}$ -linear. This gives the constant vector bundle on $\mathbb{H}(V_{I})$ with fiber $V_{I}$ a holomorphic structure (which can be identified with the total space of the dual ${\mathcal{F}}_{V_{I}}^{\vee }$ of ${\mathcal{F}}_{V_{I}}$ ). Hence, the constant vector bundle on $\mathbb{H}(V_{I})$ with fiber $V_{I}\otimes I$ also acquires a holomorphic structure (namely as the total space of ${\mathcal{F}}_{V_{I}}^{\vee }\otimes I$ ); let us denote that total space by $\mathbb{V}_{I}$ . Then $\unicode[STIX]{x1D70C}_{I}^{\prime \prime }\,:\,\mathbb{H}(V^{\prime })\rightarrow \mathbb{H}(V_{I})$ is a $\mathbb{V}_{I}$ -torsor in the complex-analytic category. A local section of $\unicode[STIX]{x1D70C}_{I}^{\prime \prime }$ identifies ${\mathcal{F}}_{V_{I}^{\prime }}$ with the pull-back along $\unicode[STIX]{x1D70C}_{I}^{\prime \prime }$ of ${\mathcal{F}}_{V_{I}}\oplus ({\mathcal{O}}_{V_{I}}\otimes I^{\ast })$ , with the group $L_{I}=\operatorname{Sp}(V_{I})\times \operatorname{GL}(I)$ acting in the obvious way. This gives the flat $\unicode[STIX]{x1D70C}_{I}^{\prime \prime }$ -connection on ${\mathcal{F}}_{V_{I}^{\prime }}$ .

The preceding also makes it clear that $P_{I}\leqslant P_{J}$ is equivalent to $I\subseteq J$ . In other words, $(\mathscr{P}_{\max },\leqslant )$ is identified with $(\mathscr{I}(V),\subseteq )$ . Note that for an inclusion $I\subseteq J$ of $\mathbb{Q}$ -isotropic subspaces, the diagram involving the associated cones is

The stable cohomology of ${\mathcal{A}}_{g}^{\text{bb}}$

We here focus on what is perhaps the most ‘classical’ example and also is a special case of Example 3.5, namely the moduli stack $\mathbb{A}_{g}$ of principally polarized abelian varieties. We shall prove that the stable cohomology of its Baily–Borel compactification contains nontrivial Tate extensions and carries Goresky–Pardon Chern classes that have nonzero imaginary part (and hence are not defined over $\mathbb{Q}$ ).

Let $H$ stand for $\mathbb{Z}^{2}$ and endowed with standard symplectic form (characterized by $\langle e,e^{\prime }\rangle =1$ , where $(e,e^{\prime })$ is its standard basis) and regard $H^{g}(=\mathbb{Z}^{2g})$ as a direct sum of symplectic lattices. In the notation of Example 3.5, we take for $V$ the vector space $\mathbb{R}\otimes H^{g}(=\mathbb{R}^{2g})$ with its obvious rational symplectic structure so that we have defined the symmetric domain $\mathbb{H}_{g}:=\mathbb{H}(\mathbb{R}\otimes H^{g})$ and we take for $\unicode[STIX]{x1D6E4}$ the integral symplectic group $\operatorname{Sp}(H^{g})(=\operatorname{Sp}(2g,\mathbb{Z}))$ . Then $\mathbb{A}_{g}$ can be identified with $\operatorname{Sp}(H^{g})\backslash \mathbb{H}_{g}$ , when we think of the latter as a Deligne–Mumford stack. The Hodge bundle on $\mathbb{H}_{g}$ descends to a rank- $g$ vector bundle ${\mathcal{F}}_{g}$ on the stack $\operatorname{Sp}(H^{g})\backslash \mathbb{H}_{g}$ . As such, it has integral Chern classes. In what follows, we will work mostly with cohomology with coefficients in $\mathbb{Q}$ -vector spaces. Then the distinction between the stack $\mathbb{A}_{g}$ and underlying coarse moduli space (that we shall denote by ${\mathcal{A}}_{g}$ ) becomes moot, for the natural map from $\mathbb{A}_{g}$ (which has the homotopy type of $B\operatorname{Sp}(2g,\mathbb{Z})$ ) to ${\mathcal{A}}_{g}$ induces an isomorphism on rational (co)homology. The Hodge bundle on $\mathbb{H}_{g}$ descends to a bundle ${\mathcal{F}}_{g}$ on $\mathbb{A}_{g}$ and thus we find $\operatorname{ch}_{k}({\mathcal{F}}_{g})\in H^{2k}(\mathbb{A}_{g};\mathbb{Q})\cong H^{2k}({\mathcal{A}}_{g};\mathbb{Q})$ . We will therefore pretend that ${\mathcal{F}}_{g}$ is a vector bundle on ${\mathcal{A}}_{g}$ . According to Charney and Lee [Reference Charney and LeeCL83], $H^{k}({\mathcal{A}}_{g}^{\text{bb}};\mathbb{Q})$ is independent of $k$ for $g$ sufficiently large. They proved that the direct sum of these stable cohomology spaces comes with the structure of a connected $\mathbb{Q}$ -Hopf algebra $H^{\bullet }$ whose primitive generators are classes $\widetilde{\text{ch}}_{2r+1}\in H^{4r+2}$ ( $r\geqslant 0$ ) and classes $y_{r}\in H^{4r+2}$ ( $r\geqslant 1$ ). For $g\gg r$ , the image of $\widetilde{\text{ch}}_{2r+1}\in H^{4r+2}$ in $H^{4r+2}({\mathcal{A}}_{g};\mathbb{Q})$ is $\operatorname{ch}_{2r+1}({\mathcal{F}}_{g})$ (which is known to be nonzero), whereas the image of $y_{r}$ in $H^{4r+2}({\mathcal{A}}_{g};\mathbb{Q})$ is zero.

The class $y_{r}$ is somewhat harder to describe: it comes from transgression of a primitive class in $H^{4r+1}(B\operatorname{GL}(\mathbb{Z});\mathbb{Q})$ , about which we will say more below. Chen and the author [Reference Chen and LooijengaJL15] have recently shown that the stability theorem holds if we take the mixed Hodge structure on $H^{\bullet }({\mathcal{A}}_{g}^{\text{bb}};\mathbb{Q})$ into account: $H^{\bullet }$ inherits such a structure with $H^{k}$ having weight ${\leqslant}k$ (for ${\mathcal{A}}_{g}^{\text{bb}}$ is compact) and $H^{k}/W_{k-1}H^{k}$ can be identified with $H^{k}({\mathcal{A}}_{g};\mathbb{Q})$ for $g$ large. For $k=4r+2$ , the image of $\widetilde{\text{ch}}_{2r+1}$ in $H^{4r+2}({\mathcal{A}}_{g};\mathbb{Q})$ is $\operatorname{ch}_{2r+1}({\mathcal{F}}_{g})$ , which is of bidegree $(2r+1,2r+1)$ (and nonzero for $g\gg r$ ), but $y_{r}$ ( $r\geqslant 1$ ) is of bidegree $(0,0)$ . So, the primitive part $H_{\text{pr}}^{4r+2}$ of $H^{4r+2}$ is for $r\geqslant 1$ a Tate extension:

(5.1) $$\begin{eqnarray}0\rightarrow \mathbb{Q}(0)\rightarrow H_{\text{pr}}^{4r+2}\rightarrow \mathbb{Q}(-2r-1)\rightarrow 0,\end{eqnarray}$$

where $\mathbb{Q}(-2r-1)$ is spanned by the image $ch_{2r+1}$ of $\widetilde{\text{ch}}_{2r+1}$ and $\mathbb{Q}(0)$ by the image of $y_{r}$ . The inclusion $\mathbb{Q}(-2r-1)\subseteq \mathbb{C}$ comes about by regarding the twisted (De Rham) version $(2\unicode[STIX]{x1D70B}\sqrt{-1})^{2r+1}\text{ch}_{2r+1}$ as the natural generator (it lies in $\mathbb{Q}(0)$ ).

In what follows, we take $g$ large enough to be in the stable range, so that this sequence appears in $H^{4r+2}({\mathcal{A}}_{g}^{\text{bb}},\mathbb{Q})$ . By Theorem 4.1 (in combination with Remarks 2.6 and 2.9), the Goresky–Pardon Chern character $\operatorname{ch}_{2r+1}^{\text{gp}}({\mathcal{F}}_{g})$ (being a universal polynomial with rational coefficients of weighted degree $2r+1$ in $\operatorname{c}_{i}^{\text{gp}}({\mathcal{F}}_{g})$ ) is then a generator of $F^{2r+1}H_{\text{pr}}^{4r+2}$ . This will help us determine the class of this extension. For this purpose, we also need to know a bit more about $y_{r}$ , when viewed as an element of $H^{4r+2}({\mathcal{A}}_{g}^{\text{bb}};\mathbb{Q})$ . We will however not describe $y_{r}$ , but rather a stable primitive homology class $z_{r}\in H_{4r+2}({\mathcal{A}}_{g}^{\text{bb}};\mathbb{Q})$ such that $\langle y_{r},z_{r}\rangle \not =0$ . That will do, for then the map $x\in H_{\text{pr}}^{4r+2}\mapsto \langle x,z_{r}\rangle /\langle y_{r},z_{r}\rangle \in \mathbb{Q}=\mathbb{Q}(0)$ splits the above sequence and so the extension class is given by the image of $\langle \operatorname{ch}_{2r+1}^{\text{gp}}({\mathcal{F}}_{g}),z_{r}\rangle$ in $\mathbb{C}/\mathbb{Q}$ . We prefer to replace $\operatorname{ch}_{2r+1}^{\text{gp}}({\mathcal{F}}_{g})$ by its De Rham variant $(2\unicode[STIX]{x1D70B}\sqrt{-1})^{2r+1}\operatorname{ch}_{2r+1}^{\text{gp}}({\mathcal{F}}_{g})$ , so that the class of this Tate extension becomes more like a period; it is then the image of $\langle (2\unicode[STIX]{x1D70B}\sqrt{-1})^{2r+1}\operatorname{ch}_{2r+1}^{\text{gp}}({\mathcal{F}}_{g}),z_{r}\rangle$ in $\mathbb{C}/\mathbb{Q}(2r+1)$ . The following theorem implies that this extension is nontrivial and that the Goresky–Pardon Chern character has a nonzero imaginary part.

The computation uses Beilinson’s regulator for the field $\mathbb{Q}$ , which involves among other things Deligne cohomology and the Cheeger–Simons classes. We recall what we need below, referring to Burgos Gil’s very accessible exposition [Reference Burgos Gilbur02] as a general reference for this topic.

Refined Chern characters

For a smooth complex variety $X$ , there is defined the Deligne cohomology group $H_{{\mathcal{D}}}^{2p}(X,\mathbb{Z}(p))$ ( $p=0,1,2,\ldots \,$ ). It fits in an exact sequence

$$\begin{eqnarray}0\rightarrow J_{p}(X)\rightarrow H_{{\mathcal{D}}}^{2p}(X;\mathbb{Z}(p))\rightarrow F^{p}H^{2p}(X;\mathbb{Z}(p))\rightarrow 0,\end{eqnarray}$$

where $F^{p}H^{2p}(X,\mathbb{Z}(p))$ denotes the intersection of the image of $H^{2p}(X;\mathbb{Z}(p))\rightarrow H^{2p}(X;\mathbb{C})$ with $F^{p}H^{2p}(X;\mathbb{C})$ and $J_{p}(X)$ is an abelian group that is the $p$ th intermediate Jacobian in case $X$ is projective:

$$\begin{eqnarray}J_{p}(X):=H^{2p-1}(X;\mathbb{C})/\big(F^{p}H^{2p-1}(X;\mathbb{C})+H^{2p-1}(X;\mathbb{Z}(p))\big).\end{eqnarray}$$

We only need here the following somewhat informal description of this extension: when $X$ is complete, an element of $H_{{\mathcal{D}}}^{2p}(X;\mathbb{Z}(p))$ is representable by a pair $(b,\unicode[STIX]{x1D6FC})$ , where $b\in H^{2p-1}(X;\mathbb{C}/\mathbb{Z}(p))$ and $\unicode[STIX]{x1D6FC}$ is a closed $2p$ -form on $X$ of Hodge level ${\geqslant}p$ with periods in $\mathbb{Z}(p)$ (we then write $\unicode[STIX]{x1D6FC}\in (F^{p}A)_{\text{cl}}^{2p}(X;\mathbb{Z}(p))$ ), such that for every smooth singular $\mathbb{Z}$ -valued $2p$ -chain $Z$ on $X$ , the image of $\int _{Z}\unicode[STIX]{x1D6FC}$ in $\mathbb{C}/\mathbb{Z}(p)$ is equal to $b([\unicode[STIX]{x2202}Z])$ . In case $X$ is not complete, we require that $\unicode[STIX]{x1D6FC}$ extends to a normal crossing compactification with logarithmic poles along $D$ of $X$ (so that it represents an element of $F^{p}H^{2p}(X)$ with periods in $\mathbb{Z}(p)$ ). The equivalence relation is the one which produces the exact sequence and so $(b,\unicode[STIX]{x1D6FC})$ represents zero precisely when the cohomology class of $\unicode[STIX]{x1D6FC}$ is zero and $b$ is in the image of $F^{p}H^{2p-1}(X;\mathbb{C})\rightarrow H^{2p-1}(X;\mathbb{C})/H^{2p-1}(X;\mathbb{Z}(p))=H^{2p-1}(X;\mathbb{C}/\mathbb{Z}(p))$ . Beilinson and Gillet showed that for a vector bundle ${\mathcal{F}}$ on $X$ one has a natural lift of $(2\unicode[STIX]{x1D70B}\sqrt{-1})^{p}\operatorname{ch}_{2p}({\mathcal{F}})\in F^{p}H^{2p}(X;\mathbb{Z}(p))$ to $H_{{\mathcal{D}}}^{2p}(X;\mathbb{Z}(p))$ . It is called the Beilinson Chern character and, in order to come to terms with the fact that Beilinson and Betti have a common initial string, we denote it by $\operatorname{ch}_{p}^{\text{B}}({\mathcal{F}})\in H_{{\mathcal{D}}}^{2p}(X;\mathbb{Z}(p))$ .