Modularity of special cycles on Shimura varieties: a survey

TL;DR

This survey reviews recent progress on Kudla's conjecture about the modularity of special cycle generating series on Shimura varieties, discussing related conjectures in broader contexts.

Contribution

It compiles and discusses recent results and formulates new conjectures on the modularity of special cycles across various Shimura varieties.

Findings

Progress on Kudla's conjecture for orthogonal and unitary Shimura varieties

Formulation of new conjectures for other Shimura varieties

Insights into the structure of special cycle classes

Abstract

We survey recent results on a conjecture of Kudla regarding the modularity of generating series of special cycle classes in toroidal compactifications of orthogonal and unitary Shimura varieties. Along the way, we formulate several conjectures on related phenomena for special cycles in other types of Shimura varieties, as well as on more general quotients of period domains.