On the global structure of special cycles on unitary Shimura varieties

TL;DR

This paper analyzes the global structure of special cycles on unitary Shimura varieties, providing explicit computations and proving their connectedness, advancing understanding of their geometric and arithmetic properties.

Contribution

It explicitly computes the structure of special cycles and their intersections on Shimura varieties, confirming a conjecture by Kudla and Rapoport.

Findings

Explicit description of the reduced loci of special cycles

Proof of connectedness of intersections of special cycles

Application of Bruhat-Tits theory to Shimura varieties

Abstract

In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for GU(1; n-1). We explicitly compute the global structure of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.