Stable cohomology of the perfect cone toroidal compactification of 𝒜 _g

Abstract

Abstract. We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification A Perf g of the moduli space of complex principally polarized abelian varieties stabilizes in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification A Matr g stabilizes in fixed degree, and forms a polynomial algebra. For degree up to 8, we describe explicitly the generators of the cohomology, and discuss various approaches to computing all of the stable cohomology in general.

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

“…The analogue of Question 1 for toroidal compactifications turns out to be a subtle question, which in this form remains open. We dealt with stability questions in a series of papers [57,58], joint with Sam Grushevsky.…”

“…The properties of the perfect cone decomposition can be rephrased in terms of the fan by saying that if a cone σ in the perfect cone decomposition has rank k, then its dimension is at least k. Moreover, the number of distinct GL(g, Z)-orbits of cones of a fixed dimension ℓ ≤ g is independent of g. This means that the combinatorics of the strata β g (σ) of codimension ℓ ≤ k is independent of g provided g ≥ k holds. Furthermore, studying the Leray spectral sequence associated to the fibration β g (σ) → A g−r with r = rank σ allows to prove that the cohomology of β g (σ) stabilizes for k < g − r − 1; this stable cohomology consists of algebraic classes and can be described explicitly in terms of the geometry of the cone σ (see [57,Theorem 8.1]). The basic idea of the proof is analogous to the one used in Theorem 21 to describe the cohomology of X n g .…”

“…In practice, however, the situation is complicated by the singularities of A Perf g . Let us review the main results of [57,58] in the case of an arbitrary sequence {A Σ g } of partial toroidal compactifications of A g associated with an admissible collection of (partial) fans Σ = {Σ g }. Here we use the word partial to stress the fact that we don't require the union of all σ ∈ Σ g to be equal to Sym 2 rc (R g ).…”

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

“…In [GHT17] we investigated the stability of cohomology of toroidal compactifications or partial toroidal compactifications of A g . The methods we used were different from the topological methods used by Borel, and Charney and Lee, and the results we obtained in [GHT17] were on stabilization in close to top degree.…”

“…In [GHT17] we investigated the stability of cohomology of toroidal compactifications or partial toroidal compactifications of A g . The methods we used were different from the topological methods used by Borel, and Charney and Lee, and the results we obtained in [GHT17] were on stabilization in close to top degree. It is easy to see that H top−k (A Sat g , Q) is independent of g for g > k (where here and below, top denotes the real dimension of the space, so in this case g(g + 1)), and is freely generated by duals of the extensions of the odd Hodge classes.…”