Slopes of Siegel cusp forms and geometry of compactified Kuga varieties

TL;DR

This paper investigates the geometric properties of compactified Kuga varieties over moduli spaces of abelian varieties, using Siegel modular forms to determine their Kodaira dimension.

Contribution

It introduces a Namikawa compactification for Kuga varieties and applies slope results of Siegel modular forms to analyze their Kodaira dimension.

Findings

Determined Kodaira dimension for all g>1 and n>0.

Constructed a Namikawa compactification with canonical singularities.

Connected modular form slopes to geometric properties of Kuga varieties.

Abstract

We study the Kodaira dimension of the compactified n-fold Kuga variety over the moduli space of principally polarised abelian g-folds. We construct a suitable compactification, which we call a Namikawa compactification, and show that in most cases it has canonical singularities. We then use results about the slope of Siegel modular forms to determine the Kodaira dimension for all g>1 and n>0.