The Kudla-Millson lift of Siegel cusp forms

TL;DR

This paper investigates the injectivity of the Kudla-Millson lift for genus 2 Siegel cusp forms, establishing conditions under which the lift is injective and exploring its implications for cohomology of orthogonal Shimura varieties.

Contribution

It proves the injectivity of the Kudla-Millson lift for certain lattices and introduces vector-valued indefinite Siegel theta functions, extending Borcherds' results to higher genus.

Findings

Lift is injective when L splits off two hyperbolic planes and has large rank.

The image of the lift matches the dimension of the space of cusp forms in cohomology.

Results apply to moduli spaces of quasi-polarized K3 surfaces.

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 results of Borcherds to genus higher than 1.