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

The relative proportionality principle of Hirzebruch and Höfer was discovered in the case of compactified ball quotient surfaces X when studying curves C⊂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 of type M=Γ SO(n,2)/K. 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 Wi which are of Hodge type and satisfy relative proportionality.