On the connectedness of the singular locus of the moduli space of principally polarized abelian varieties

TL;DR

This paper proves that the subset of the moduli space of principally polarized abelian varieties, characterized by the presence of a specific involution, forms a connected sublocus.

Contribution

It establishes the connectedness of the singular locus in the moduli space related to abelian varieties with a particular involution, a new topological insight.

Findings

The singular locus with involutions is connected.

The result applies for dimensions g ≥ 3.

Enhances understanding of the moduli space topology.

Abstract

Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.