Derived special cycles on Shimura varieties

TL;DR

This paper uses derived algebraic geometry to construct special cycle classes on integral models of various Shimura varieties, providing a unified approach that supports Kudla's modularity conjectures.

Contribution

It introduces a uniform moduli-theoretic construction of special cycles on Shimura varieties of Hodge type using derived algebraic geometry, extending to degenerate cases.

Findings

Construction of special cycle classes with desired properties

Formulation of Kudla's modularity conjectures in this framework

Preliminary evidence supporting the conjectures

Abstract

I employ methods from derived algebraic geometry to give a uniform moduli-theoretic construction of special cycle classes on integral models many Shimura varieties of Hodge type, including unitary, quaternionic, and orthogonal Shimura varieties. All desired properties of these cycles, even for those corresponding to degenerate Fourier coefficients under the Kudla correspondence, follow naturally from the construction. I formulate Kudla's modularity conjectures in this general framework, and give some preliminary evidence towards their validity.