Goresky–Pardon lifts of Chern classes and associated Tate extensions

Abstract

Let X be an irreducible complex-analytic variety, S a stratification of X and F a holomorphic vector bundle on the open stratumX. We give geometric conditions on S and F that produce a natural lift of the Chern class c k (F ) ∈ H 2k (X; C) to H 2k (X; C), which, in the algebraic setting, is of Hodge level ≥ k. When applied to the Baily-Borel compactification X of a locally symmetric varietẙ X and an automorphic vector bundle F onX, this refines a theorem of Goresky-Pardon. In passing we define a class of simplicial resolutions of the Baily-Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake (=Baily-Borel) compactification of Ag contains nontrivial Tate extensions.

The stable cohomology of the Satake compactification of 𝒜g

“…We will use the fact that the higher direct images R • j g * Q Ag are locally constant on each stratum A r . Each point of A r has a neighborhood basis whose members meet A g in a virtual classifying space of an arithmetic group P g (r) defined below (for a more detailed discussion we refer to [11], Example 3.5; see also Section 4 of [5]), so that R • j g * Q Ag can be identified with the rational cohomology of P g (r).…”

“…Goresky and Pardon [10] have constructed a lift c bb r of the real Chern class c r ∈ H 2r (A g ; R) to H 2r (A bb g ; R). The second author [11] recently proved that c bb r (and hence also the corresponding Chern character ch bb r ) lies in F r H 2r (A bb g ; R). So the class of the Tate extension in Remark 1.3 is up to a rational number given by the value of c bb 2r+1 on the class z r ∈ H 4r+2 (A bb g ; Q) found in Remark 2.5 (two choices of z r differ by a class of the form j g * (w) with w ∈ H 4r+2 (A g ; Q) and c bb 2r+1 takes on such a class the rational value c 2r+1 (w)).…”

“…Arvind Nair, after learning of our theorem, informed us that his techniques enable him to show that this extension class is nonzero. Subsequently a different proof (based on the Beilinson regulator) was given in [11].…”

The intersection cohomology of the Satake compactification of $${\mathcal {A}}_g$$ for $$g \le 4$$

“…We remark that another lift of the classes λ i to A Sat g was constructed by Goresky and Pardon [GP], but their lifts a priori only live in cohomology H 2i (A Sat g , C) with complex coefficients. In fact, by a result of Looijenga [Lo2] the Goresky-Pardon classes have a non-trivial imaginary part and thus do not live in H 2i (A Sat g , Q). Since, however, we prefer to work with rational coefficients, we will work with Charney-Lee classes.…”

Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

“…Chen and Looijenga proved in [CL16] that these classes have Hodge weight zero, and thus are non-algebraic. Thus the stable cohomology of A Sat g contains non-trivial Tate extensions, and we refer the reader to the recent preprint [Loo15] for a discussion of these extensions.…”