Relative Proportionality for subvarieties of moduli spaces of K3 and abelian surfaces

TL;DR

This paper generalizes the relative proportionality principle from surfaces to higher-dimensional Shimura varieties of orthogonal type, providing criteria to identify Hodge-type subvarieties based on divisors satisfying proportionality conditions.

Contribution

It extends the relative proportionality inequality to subvarieties of Shimura varieties of orthogonal type and explores conditions for subvarieties to be of Hodge type.

Findings

Proved a generalized proportionality inequality for subvarieties of orthogonal Shimura varieties.

Established criteria for identifying Hodge-type subvarieties based on divisors satisfying proportionality.

Enhanced understanding of the geometric structure of subvarieties in Shimura varieties.

Abstract

The relative proportionality principle of Hirzebruch and H\"ofer was discovered in the case of compactified ball quotient surfaces X when studying curves C in X. It can be expressed as an inequality which attains equality precisely when C is an induced quotient of a subball. A similar inequality holds for curves on Hilbert modular surfaces. In this paper we prove a generalization of this result to subvarieties of Shimura varieties of orthogonal type, i.e. locally symmetric spaces for the Lie group SO(n,2). Furthermore we study the ''inverse problem'' of deciding when an arbitrary subvariety Z of M is of Hodge type, provided it contains sufficiently many divisors W_i which are of Hodge type and satisfy relative proportionality.