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

TL;DR

This paper computes the intersection cohomology of the Satake compactification of the moduli space of principally polarized abelian varieties for genera 2, 3, and 4, providing detailed structural insights and decomposition results.

Contribution

It fully determines the intersection cohomology for these compactifications in low genera, advancing understanding of their geometric and topological structure.

Findings

Complete intersection cohomology calculations for genus 2, 3, 4

Decomposition theorem ingredients identified for key maps

Results on intersection cohomology of link bundles

Abstract

We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 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.