Arithmetic special cycles on unitary Shimura varieties and Eisenstein series

I will discuss some aspects of the theory of 'special' cycles on PEL Shimura varieties.
The relationship between generating series built out of such cycles on the one hand, and the Fourier expansions of modular forms on the other, is a modern take on an old subject. Over a number field, these relations can be explained as geometric versions of the theory of theta series going back to Siegel and Weil. The situation when one considers integral models is rather more subtle: deep conjectures due to Kudla suggest that generating series of arithmetic cycles should be related to special values of derivatives of Eisenstein series.

Recent work of Kudla and Rapoport describes an approach to these conjectures in the context of unitary Shimura varieties of signature (n,1). In the talk I will describe in broad terms some of the attendant problems that arise, including the relation to cycles on Rapoport-Zink spaces, and survey the known results. Time permitting, I will also illustrate how some of these issues are resolved in the case of 0-cycles on certain unitary Shimura varieties of signature (1,1).