# Lifting Chern classes by means of Ekedahl-Oort strata

###### Abstract.

The moduli space of principally polarized abelian varieties of genus is defined over and admits a minimal compactification , also defined over . The Hodge bundle over has its Chern classes in the Chow ring of with -coefficients. We show that over , these Chern classes naturally lift to and do so in the best possible way: despite the highly singular nature of they are represented by algebraic cycles on which define elements in the bivariant Chow ring. This is in contrast to the situation in the analytoc topology, where these Chern classes have canonical lifts to complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

###### 1991 Mathematics Subject Classification:

14## 1. Introduction and statement of the main result

Few objects in algebraic geometry have such a rich structure as the moduli space of principally polarized
abelian varieties of dimension . Its modular interpretation
makes it a stack over and it comes as such with a rank vector bundle, the *Hodge bundle*
(which we may regard as the basic automorphic bundle over
in the sense that all other such over
are manufactured from it). Its determinant bundle is ample and when , the graded algebra of automorphic forms
is finitely generated so that its Proj defines a natural projective completion of .
The complex-analytic space
underlying has the familiar description as the quotient of the Siegel upper half space of genus
by the integral symplectic group and is then the Satake-Baily-Borel compactification.
Since is a Deligne-Mumford stack, the (operational) Chow ring of
with -coefficients, is well-defined.
The Chern classes of generate a subalgebra herein (we recall its presentation below). Since every automorphic bundle over is universally expressed via a Schur functor
in terms of its Hodge bundle, contains the Chern classes of all such bundles. This is why we refer to as the *tautological ring* of

The cycle map embeds this ring in
and its image can be characterized in several ways. One
is that after tensoring with , it is the subalgebra representable by differential forms whose pull-backs to
are -invariant (such forms are automatically closed).
Charney and Lee [3] had shown in 1983 that in the stable range (that is, for cohomological degree ) these classes are liftable to , but Goresky and Pardon [14] proved in 2002
that they admit in fact a *natural* lift, provided we use the *complex* cohomology of .
They raised the question whether their lifts lie in the rational cohomology. The answer to that question, given by one of us [18],
is in general no. To see why, it is better to use the Chern characters rather than the Chern classes, for then the even indexed Chern characters are zero, so that the issue regarding liftability only concerns the odd indexed ones. The answer is then that for odd, the Goresky-Pardon lift of lies in the Hodge space . If we are also in the stable range , then, as we recall below, it lies in fact in the complexification of a mixed Tate substructure of : an extension of by . This extension is nontrivial
in the sense that it is proportional to a standard nontrivial one whose invariant is given by . Since is odd, this implies that in this range,
will not even be a real cohomology class.

We noted already back in 2015 that the situation is entirely different for . For this, let us recall that Ekedahl and van der Geer [4] had proved that is then generated by the Ekedahl-Oort strata. Our observation at the time was that these strata intersect the boundary of transversally with respect to its natural stratification (with “minimal perversity”), which means that these classes naturally lift to -adic cohomology classes on . We then realized that the notion of an -zip, introduced by Moonen and Wedhorn in [19] and the classifying space of such as introduced by Pink-Wedhorn-Ziegler [22] make it fit into an even neater picture. This classifying space of zips is an Artin stack, denoted (we give more details below), which can be regarded as the characteristic counterpart of the compact dual of the Siegel upper space . The Chow ring is isomorphic to the one of . We have a natural morphism of Artin stacks . It has the property that it maps onto . Our main observation now becomes:

###### Theorem 1.1.

The morphism naturally extends to the minimal compactification: and the induced ring homomorphism is an embedding.

Here the ring is Fulton’s bivariant Chow ring [9]. One may be tempted to call this image the tautological ring of , although (as was shown in [3]), the stable cohomology of the Baily-Borel compactification is larger than the algebra generated by the .

###### Remark 1.2.

A natural analogue of this theorem can be stated for Shimura varieties of Hodge type, where the role of is taken by the subalgebra , with a finite field, generated by Chern classes of automorphic bundles. Here the subscript refers to the algebraic group that is part of the data that give the structure of a Shimura variety. It is here implicit that we employ an integral model for which has good reduction over the prime with residue field . Such models have been constructed by Vasiu [26] and Kisin [15].

The work of Pink, Wedhorn and Ziegler [22, 23] applies to this setting: we still have a moduli stack of zips and a classifying morphism , the fibres of which are the Ekedahl-Oort strata. The Chow -algebra of (here denoted ) is according to [2, Thm. 2.4.4] isomorphic to that of the compact dual . If is faithfully flat and surjective and can be extended to a morphism of a toroidal compactification of Faltings-Chai type, then essentially the same proof shows that embeds in the Chow algebra of the toroidal compactification (see [27, 29, 20, 16] for results in this direction). The strata extend to the boundary and enjoy good intersection properties with the boundary, see [1, Thm. 6.1.6] and [17]. The morphism factors through a morphism of the minimal compactification to the stack and we thus find in a way similar to the case of a copy of in the Chow algebra of the minimal compactification . We will confine ourselves however to the case .

Let us note that Esnault and Harris [6] recently proved a lifting property in the case of mixed characteristic, but on the level of -adic cohomology. It would be interesting to see whether their result can be lifted to the level of Chow algebras.

## 2. The Case

### 2.1. Review of the situation in characteristic zero.

We let be a toroidal compactification of of Faltings-Chai type and denote by the natural projection. The Hodge bundle on extends to and this extension is again denoted by .

The analytic space of the complex fibre can be described in terms of the Chevalley group , the automorphism group of the standard symplectic lattice as where is a bounded symmetric domain with a maximal compact subgroup.

Let us briefly review what is known about the Chow ring of the compact dual of in the more general case where is a reductive algebraic -group whose symmetric space has the structure of a bounded symmetric domain. Then the compact dual of is of the form with a maximal parabolic subgroup of . We have a decomposition into Schubert cells: , where runs over the elements of the Weyl group of , or rather (in order to keep the union disjoint), over a complete set of coset representatives for , where is the subgroup of associated to . It is known that the Chow ring has as an additive basis the classes of the closures of Schubert cells (Schubert varieties) in . The ring structure on the Chow ring with -coefficients, , is described by Borel (see [25, p. 142, (28)]:

Here is the symmetric -algebra on the character group of a Borel subgroup, is the invariant part under and is the ideal generated by -invariant elements of positive degree. In case the group is ‘special’, e.g. for and , this isomorphism also holds for -coefficients.

In our case, where , this graded -algebra is isomorphic to

where has degree and is the ideal generated by the graded pieces of

So this gives a relation in every positive even degree . Note that .

For a field , the Chern classes in satisfy the same relation as the in the Chow ring of as the :

(see [10, 7]) and they generate a subring of the Chow ring isomorphic to the rational Chow ring of . This extends the Hirzebruch-Mumford Proportionality to the Chow rings. This ring is called the tautological subring of and denoted again by . Its image in under restriction via is .

### 2.2. The Artin stack of zips.

We now restrict to characteristic and consider
and
.
The compact dual of Siegel space (or of any symmetric domain) has no
obvious counterpart in positive characteristic.
But it turns out that there is a good substitute, *viz.* the Artin stack of zips, that can take on that role
for our purposes. Its origin is the
so-called Ekedahl-Oort stratification, introduced in [21].
As we will recall below, there are strata
and the cycle classes
of the (closed) strata lie in the tautological subring,
as shown in [10, 4].
As examples we have the (closed) -rank loci (-rank
with )
with cycle classes .
Thus the generators of
are represented by these effective cycles.

#### The basic definition

For a principally polarized abelian variety of dimension
in characteristic the de Rham cohomology space
comes equipped with a non-degenerate alternating form.
The Frobenius operator induces a -linear endomorphism of whose kernel is
its Hodge subspace . Both the kernel and the
image of this endomorphism are Lagrangian subspaces of dimension .
As we will see below, this structure (consisting of a symplectic vector space and a Frobenius-linear
endomorphism of
whose kernel and image is a Lagrangian subspace) has only
finitely many isomorphism types. Such a structure
is called a *zip* and was studied in [19].
(To make the isomorphism type explicit one usually endows
kernel and image with filtrations by taking preimages
and images of iterates of , and then extends these
to self-dual filtrations on by adding their symplectic perps.
This results in a descending filtration
(a refinement of the *Hodge filtration*) ,
and an ascending filtration (a refinement of
the *conjugate filtration*)
, connected by the Cartier operator giving Frobenius-linear
identifications . The dimensions
of the intersections of these filtrations determine the isomorphism type.
This will however not matter to us in what follows.)

#### Moduli space and Schubert varieties

In an evident manner we have defined a moduli space of all zip structures on :
if is the Grassmannian of Lagrangian subspaces of and denotes its universal bundle, then
is an open subset in the total space of the exterior tensor product bundle
over . The group acts in an evident manner on . We shall call the closure of a -orbit in a *Schubert variety*.

There are such Schubert varieties. This is based on the observation that the relative position of a pair of Lagrangian subspaces (in other words, the -orbit of such a pair) is given by a double coset of : if (resp. ) is the -stabilizer of (resp. ), then the for which make up the double coset , so that we get an element of . We can identify this set of double cosets in terms of Weyl groups: if we choose a Borel subgroup contained in with maximal torus and (resp. ) is the normalizer of in (resp. in ), then (resp. is ) is the Weyl group of the pair (resp. ) and it is a standard fact of the theory of algebraic groups that the natural map

is a bijection. One finds that in our case has elements, and hence as many Schubert varieties.

#### The Artin stack of zip data

Let us make here the connection with the way this notion appears in the literature. The groups and are maximal parabolic subgroups of
whose Levi quotients resp. can be identified
with the general linear groups of (or of its dual for that matter) resp. . So an isomorphism can be understood as giving an isomorphism up to a scalar. Similarly, a Frobenius-linear map of onto determines a Frobenius isogeny
.
We can formulate this in terms of only: in our setting a *zip datum* is given by a -tuple , where
, and are maximal parabolic subgroups
of and is an isogeny between their Levi quotients given by Frobenius.
We form the fibre product of and over (the former via the group homomorphism ) in the category
of algebraic groups:

This group acts on by and we can form the Artin stack . Brokemper determined the Chow ring of the stack (which is essentially by definition the -equivariant Chow ring of ). He considers in [2] more generally the case of a connected group and an algebraic zip datum. Choose such that . If we identify resp. with their images in resp. , then we can even arrange that takes to , so that we have defined an isogeny

Then acts on , the symmetric algebra of , the character group of . The Chow ring of the stack is ([2, Thm. 2.4.4, page 27]

In our case, this group is additively generated by the Schubert varieties as defined above.

This Chow ring can be regarded as the ring of characteristic classes for symplectic vector bundles over endowed with a zip structure.
For if have a symplectic vector bundle over a base scheme (or stack, for that matter) over of rank , then the above construction yields the zip bundle over , so that to endow
with a zip structure amounts to giving a section of . This comes with relative Schubert varieties and these define an
embedding of in Fulton’s bivariant Chow ring as a subalgebra, having these relative Schubert varieties as additive generators.
If a zip structure on has associated section , then we may define its ring of characteristic classes as the image of this subalgebra under
. Note that when has proper intersection with a given relative Schubert variety in , then the associated class is represented by a *specific algebraic cycle * on defined over ; we shall refer to these as the *Ekedahl-Oort cycles*.

### 2.3. Degenerations of zips

Let us for a moment return to our fixed symplectic vector space over and suppose we are given an isotropic subspace over . Then is a symplectic vector space over and we if assign to a Lagrangian subspace which contains the subspace , we get a bijection between the Lagrangian subspaces of containing and the Lagrangian subspaces of . Denote by the subscheme defined by the Frobenius-linear endomorphisms of that are zero on , preserve , and induce the Frobenius on . The kernel of is sandwiched between and and the induced endomorphism of defines an element of , as both its kernel and image are Lagrangian subspaces. The resulting morphism is equivariant over the evident group homomorphism from the -stabilizer of to and this makes a torsor over a vector bundle on . The preimage of a Schubert subvariety of is contained in a Schubert subvariety of of the same codimension. To be precise, every -orbit in orbit meets transversally, and when this intersection is nonempty, then it is the preimage a -orbit in .

We use these observations to understand a class of degenerations of zips over a discrete valuation ring. Let be a discrete valuation ring of finite type over with residue field and field of fractions . Assume that we are given a symplectic -vector space of rank , an isotropic -subspace (so that is symplectic -vector space) and an -lattice . If denotes the fiber over the closed point, then we have an evident specialization map .

###### Lemma 2.1.

Let be such that the image of in lands in and specializes over the closed point to , say. Assume that has proper intersections with the Schubert varieties over . Then has proper intersections with the Schubert varieties over and the Ekedahl-Oort cycles of on extend to and specialize to the Ekedahl-Oort cycles of .

The proof (which we omit) is essentially a linear algebra exercise, as it amounts to finding a -invariant -lattice which induces the given -lattice of and is such that .

### 2.4. Extension of the stratification across the Satake compactification

By assigning to a principally polarized abelian variety
of dimension the isomorphism type of its zip
on its first de Rham cohomology space, we obtain a stratification
of the moduli space , the *Ekedahl-Oort stratification*. It is
is induced by a morphism of stacks

This morphism is smooth (see [29, Thm. 4.1.2]) and the fibres are the strata.

This stratification can be extended to a toroidal compactification (of Chai-Faltings type) . The space admits a stratification by torus rank: if is the canonical map to the Baily-Borel compactification and is the standard decomposition, then the restriction of the Hodge bundle to , contains a rank subbundle which is the pullback of the Hodge bundle on .

The canonical extension of the de Rham complex is the logarithmic de Rham complex where logarithmic singularities along the divisor added to compactify the semi-abelian variety are allowed, cf. [8, VI, Theorem 1.1, p. 195]. The logarithmic de Rham sheaf

extends the de Rham sheaf . On it comes again with two filtrations forming a zip. In fact, the morphism can be extended to a morphism which is again smooth as can be seen by using [4, Lemma 5.1] or [1], see also below. The closed strata on are the closures of the strata on .

The Ekedahl-Oort stratification on intersects the boundary strata transversally as we will now explain. The reason is that the Ekedahl-Oort stratification is defined by the action of Frobenius and Verschiebung acting on the logarithmic de Rham cohomology of a semi-abelian variety and on the toric part this action is essentially trivial.

We consider a semi-abelian variety over with a discrete valuation ring of finite type over . We assume that the generic fibre is abelian and the special fibre is the Néron model of a semi-abelian variety of torus rank . We let be a toroidal compactification of of Faltings-Chai type. It can be obtained via the action on a semi-abelian variety over by a group of periods with free abelian of rank . Here the semi-abelian variety is an extension of an abelian scheme by a split torus of rank . In this case the logarithmic de Rham cohomology can be described with the help of universal vector extensions, that is, extensions of group schemes by vector group schemes. We refer to [8] pages 81–86 for a description. The universal vector extension of is a vector group extension

that is canonically isomorphic to the pullback under of the universal vector extension of , where is the sheaf of invariant relative -forms on the dual abelian variety of . For the quotient construction we need an equivariant form of this, that is, we need in addition a lifting of the homomorphism to . Then acts via translation.

The dual of the logarithmic de Rham cohomology is the -equivariant Lie-algebra of the universal vector extension of . It contains the Lie algebra of as subspace. The valuation on defines a -valued bilinear form on which we can see as the analogue of the monodromy operator of Hodge theory. Its invariant part defines a subspace of dimension in the special fibre of the logarithmic de Rham cohomology over . (One might view it as the Dieudonné module of the kernel of multiplication by on the semi-abelian special fibre of .) Frobenius acts with trivial kernel on and is contained in the image. We are thus in the situation described above in subsection 2.3. Since the kernel of Frobenius has zero intersection with the isomorphism type of the zip on the special fibre of depends only on the zip of the de Rham cohomology of the abelian part. We can apply Lemma 2.1 to conclude that the closures of the strata on are the strata on and by induction that the intersection with the boundary strata is proper.

We thus see that the map factors through a map

The morphism induces a homomorphism of Chow rings

and it induces an isomorphism . Indeed, the closed Ekedahl-Oort strata on are effective cycles with non-zero classes.

###### Proof of Theorem 1.1.

The image under push forward via of in the Chow cohomology group is independent of the chosen toroidal compactification, see [5, Def-Prop. 3.1]. Thus these define classes in . On the other hand we have the generators of the Chow ring of the stack and via the map these act as bivariant classes by cap product on the Chow groups of . These satisfy . By [9, 17.1] and the projection formula ([9, p. 323]) we have

for all . This enables us identify the bivariant classes with the . It thus gives rise to a diagram

∎

###### Remark 2.2.

In the end the argument is based on the observation that all the tautological classes have an effective representative on that intersects the boundary properly. This fails to be so in characteristic zero, although it is then true for the ample , and hence for any power on , like . But this is not so for . This seems related to the question of whether for a given field the space contains complete subvarieties of codimension . For every complete subvariety of has as class a multiple of . Conversely, an effective representative for transversal to the boundary of does not intersect the boundary because , hence yields a complete subvariety of codimension .

## Acknowledgements

We thank Luc Illusie and Gerd Faltings for correspondence.

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