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

Abstract

We completely determine the intersection cohomology of the Satake compactifications A Sat 2 , A Sat 3 , and A Sat 4 , except for IH 10 (A Sat 4 ). We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.

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

“…Finally we notice the following connection between the tautological ring R g and intersection cohomology of A Sat g , see also [56]:…”

“…For g ≤ 11 and "small" λ, the classification by Chenevier and Lannes in [23] of level one algebraic automorphic representations of general linear groups over Q having "motivic weight" ≤ 22 (see Theorem 31 below) gives another method to compute these sets. Using either method, we deduce IH 6 (A Sat 3 , V 1,1,0 ) = 0 in Corollary 3, which was a missing ingredient to complete the computation in [56] of IH • (A Sat 4 , Q) (case g = 4 in Theorem 17). In fact using the computation by Vogan and Zuckerman [102] of the (g, K)-cohomology of Adams-Johnson representations, including the trivial representation of Sp 2g (R), we can prove that the intersection cohomology of A Sat g is isomorphic to the tautological ring R g for all g ≤ 5 (see Theorem 17), again by either method.…”

“…This calculation works surprisingly well and even provides information on the vanishing of the intersection cohomology of certain links and local systems. The missing piece of information in [56] was the intersection cohomology group IH 6 (A Sat 3 , V 11 ) which indeed vanishes, see Corollary (3) in the appendix.…”

“…We want to conclude this section with a discussion about the intersection cohomology of A Sat g and the toroidal compactifications for small genus. For g ≤ 4 a geometric approach was given in [56] where all intersection Betti numbers with the exception of the middle Betti number in genus 4 were determined. As it was pointed out to us by Dan Petersen, representation theoretic methods and in particular the work by O. Taïbi, give an alternative and very powerful method for computing these numbers, as will be discussed in the appendix, in particular Theorem 32.…”

“…we shall first discuss the geometric approach. This was developed in [56] and is based on the decomposition theorem due to Beilinson, Bernstein, Deligne and Gabber. For this we refer the reader to the excellent survey paper by de Cataldo and Migliorini [27].…”