Cones of special cycles and unfolding of the Kudla-Millson lift

This PhD thesis, divided in 4 chapters, sheds further light on the geometric and arithmetic properties of orthogonal Shimura varieties. In the first chapter, we describe the cone C(X) generated by special cycles of codimension 2 on an orthogonal Shimura variety X. More precisely, we prove that the accumulation cone of C(X) is pointed, rational and polyhedral. The idea is to show analogous properties for the cones of Fourier coefficients of Siegel modular forms. We also compute the accumulation rays of C(X), proving that they are generated by combinations of Heegner divisors intersected with the Hodge class of X. Eventually, we conjecture the polyhedrality of C(X), translating it into properties of Fourier coefficients of Jacobi cusp forms. We now describe the content of the second chapter. Consider a sequence of pairwise different orthogonal Shimura subvarieties of fixed dimension r>2 in X. We prove that there exists a subsequence and an orthogonal Shimura subvariety Z of X, such that the subvarieties in the subsequence equidistribute in Z. We then compute the limits of the sequence of normalized cohomology classes. Eventually, we explain a strategy to compute the accumulation rays of the cones generated by special cycles on X via the previous results. The goal of Chapter 3 is to unfold the defining integrals of the Kudla–Millson lift of genus 1, associated to even lattices of signature (b,2), where b>2. This enables us to compute the Fourier expansion of such defining integrals. As application, we prove the injectivity of the Kudla–Millson lift. Although this was already proved by Bruinier, our procedure has the advantage of paving the ground for a strategy that can work for the case of genus greater than 1. In Chapter 4 we apply a similar strategy as in Chapter 3 to the genus 2 case. We unfold the defining integrals of the Kudla–Millson lift of genus 2, under the condition that the latter is associated to some even unimodular lattice of signature (b,2), where b>2. We explain why this unfolding is not enough to prove the injectivity of the lift, showing why an additional unfolding of integrals of Jacobi type seems necessary.