Modularity of Special Cycles in Orthogonal and Unitary Shimura Varieties

Explore an advanced mathematical lecture examining the modularity properties of special cycles within orthogonal and unitary Shimura varieties. Delve into the historical foundations established by Jacobi and Siegel regarding theta series of quadratic lattices and their production of modular forms. Understand the groundbreaking generalization by Kudla and Millson, which demonstrated that generating series of special cycles in orthogonal and unitary Shimura varieties constitute modular forms. Learn about recent extensions of these fundamental results to toroidal compactifications, where special cycles are corrected by specific boundary cycles to maintain modularity properties. Examine the proof that these corrected generating series remain modular forms for divisors in orthogonal Shimura varieties and for cycles of codimension up to the middle degree in the cohomology of unitary Shimura varieties. Discover how these findings provide partial answers to Kudla's conjecture, advancing our understanding of the deep connections between algebraic geometry, number theory, and modular forms. Gain insights into the sophisticated mathematical techniques required to handle boundary corrections in compactified Shimura varieties and their implications for the broader theory of special cycles.