The stable cohomology of the Satake compactification of 𝒜g

Abstract

ABSTRACT. Charney and Lee [7] have shown that the rational cohomology of the Satake-Baily-Borel compactification A bb g of Ag stabilizes as g → ∞ and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebrogeometric proof of their theorem and show that this stable cohomology comes with a mixed Hodge structure of which we determine the Hodge numbers. We find that the mixed Hodge structure on the primitive cohomology in degrees 4r + 2 with r ≥ 1 is an extension of Q(−2r − 1) by Q(0); in particular, it is not pure.

Stable cohomology of the perfect cone toroidal compactification of 𝒜 g

“…By the results of [Hai02] there is a non-algebraic class (it has a wrong Tate twist) in H 6 (A Sat 3 ), which is likely to be α 3 , and it follows from the results of [HT12] that there is also a non-algebraic class in H 8 (A Sat 4 ), which is likely to be α 3 λ 1 . Furthermore, Chen and Looijenga [ChL15] recently proved that all α i are of Hodge type (0, 0), which in particular implies that they are not algebraic.…”

The Topology of A g $${{{\mathcal A}}_{g}}$$ and Its Compactifications

“…However, because of the fact that Charney and Lee replace A Sat g with its Q-homology equivalent space |W g |, the geometric meaning of the x-and y-classes remains unclear. This gives rise to the following two questions: [22]. Basically, in their paper they succeed in redoing Charney-Lee's proof using only algebro-geometric constructions.…”

“…If one considers the Hopf algebra structure of stable cohomology of A Sat g , the primitive part in degree 4r + 2 is generated by y 4r+2 and the Goresky-Pardon lift ch GP 2r+1 of the Chern character, which is a degree 2r + 1 polynomial in the λ GP j with j ≤ 2r + 1. The proof of Theorem 24 is based on an explicit computation of the Hodge structures on this primitive part ([78, Theorem 5.1]) obtained by using the theory of Beilinson regulator and the explicit description of the y-generators given in [22]. This amounts to describing H 4r+2 (A Sat g , Q) prim as an extension of a weight 4r + 2 Hodge structure generated by ch GP 2r+1 by a weight 0 Hodge structure, generated by y 4r+2 .…”

“…[22, Theorem 1.2]). The y-classes have Hodge type (0, 0).Concerning Question 3, it is clear that while the y-classes are canonically defined, the x-classes are not.…”

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.…”