The Kudla–Millson lift of Siegel cusp forms

ABSTRACT We study the injectivity of the Kudla–Millson lift of genus-2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L . We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree-4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L . We describe their behavior with respect to the split of a hyperbolic plane in L . This generalizes the results of Borcherds to genus higher than 1.