Siegel operators for holomorphic differential forms

TL;DR

This paper provides a geometric interpretation of Siegel operators for holomorphic differential forms on Siegel modular varieties, linking boundary behavior, compactification, and orthogonal modular varieties.

Contribution

It offers a new geometric perspective on Siegel operators, extending differential forms over compactifications and relating boundary restrictions to modular forms.

Findings

Extension of differential forms over toroidal compactification

Equivalence of boundary vanishing notions for holomorphic forms

Application to orthogonal modular varieties

Abstract

We give a geometric interpretation of the Siegel operators for holomorphic differential forms on Siegel modular varieties. This involves extension of the differential forms over a toroidal compactification, and we show that the Siegel operator essentially describes the restriction and descent to the boundary Kuga variety via holomorphic Leray filtration. As a consequence, we obtain equivalence of various notions of "vanishing at boundary'' for holomorphic forms. We also study the case of orthogonal modular varieties.