Modularity of generating functions of special cycles on Shimura varieties

Abstract

In this thesis we study special cycles on Shimura varieties of orthogonal type. We confirm a conjecture of Kudla in [K2] on the modularity of generating functions of special cycles of any codimension on Shimura varieties of orthogonal type, provided their convergence. This is a generalization of theorems of Hirzebruch-Zagier, Gross-Kohnen-Zagier and Borcherds to high codimensional cycles.