Normal functions over locally symmetric varieties

An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data. In a variational context, the resulting section of a bundle of complex tori is called a normal function, and is used to study cycles modulo rational or algebraic equivalence.
The archetype for interesting normal functions arises from the Ceresa cycle, consisting of the difference of two copies of a curve in its Jacobian. The profound geometric consequences of its existence are evidenced in work of Nori, Hain and (most recently) Totaro. In contrast, a theorem of Green and Voisin demonstrates the absence of normal functions arising from cycles on very general projective hypersurfaces of large enough degree.
Inspired by recent work of Friedman-Laza on Hermitian variation of Hodge structure and Oort's conjecture on special subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over étale neighborhoods of Shimura varieties. In this talk I will explain our classification of the cases where a Green-Voisin analogue does not hold, and where one expects interesting cycles (and generalized cycles) to occur. I will also give evidence that these predictions might be "sharp", and draw some geometric consequences.