A characterization of special subvarieties in orthogonal Shimura varieties

TL;DR

This paper establishes criteria for identifying when a subvariety in an orthogonal Shimura variety's compactification is itself a special subvariety, extending André-Oort type results to boundary-intersecting cases.

Contribution

It provides necessary and sufficient conditions characterizing special subvarieties in orthogonal Shimura varieties that intersect the boundary transversally.

Findings

Criteria for special subvarieties in compactified orthogonal Shimura varieties.

Extension of André-Oort type conditions to boundary cases.

Characterization of subvarieties intersecting the boundary transversally.

Abstract

Let $Y$ be a subvariety contained in a smooth Mumford compactification of an orthogonal Shimura variety $M \subset A_g$, where $A_g$ is the moduli space of principally polarized abelian varieties of dimension $g$ with some level structure, such that $Y$ intersects the boundary of $A_g$ transversally. Then we give necessary and sufficient conditions of Andr\'e-Oort type for $Y$ itself being the compactification of a special subvariety $Y^0 \subset M$