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We study the injectivity of the Kudla-Millson lift of genus 2 Siegel cuspforms, vector-valued with respect to the Weil representation associated to aneven lattice L. We prove that if L splits off two hyperbolic planes and is ofsufficiently large rank, then the lift is injective. As an application, wededuce that the image of the lift in the degree 4 cohomology of the associatedorthogonal Shimura variety has the same dimension as the lifted space of cuspforms. Our results also cover the case of moduli spaces of quasi-polarized K3surfaces. To prove the injectivity, we introduce vector-valued indefiniteSiegel theta functions of genus 2 and of Jacobi type attached to L. We describetheir behavior with respect to the split of a hyperbolic pane in L. Thisgeneralizes results of Borcherds to genus higher than 1.

The Kudla-Millson lift of Siegel cusp forms