Derived special cycles on Shimura varieties

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.