Lefschetz decompositions of Kudla-Millson theta functions

TL;DR

This paper explores the Lefschetz decomposition of Kudla-Millson theta functions, revealing their modular structure and implications for the cohomology of orthogonal Shimura varieties, including moduli spaces of hyperkähler manifolds.

Contribution

It establishes that the Lefschetz decomposition aligns with the modular decomposition of Kudla-Millson theta functions and shows the lift's image lies in primitive cohomology, with applications to cohomology dimension formulas.

Findings

Lefschetz decomposition matches modular decomposition into Eisenstein, Klingen, and cuspidal parts.

The Kudla-Millson lift maps into primitive cohomology.

Derived dimension formulas for cohomology in low degrees.

Abstract

In the 80's Kudla and Millson introduced a theta function in two variables. It behaves as a Siegel modular form with respect to the first variable, and is a closed differential form on an orthogonal Shimura variety with respect to the other variable. We prove that the Lefschetz decomposition of the cohomology class of that theta function is also its modular decomposition in Eisenstein, Klingen and cuspidal parts. We also show that the image of the Kudla-Millson lift is contained in the primitive cohomology of X. As an application, we deduce dimension formulas for the cohomology of X in low degree. Our results cover the cases of moduli spaces of compact hyperk\"ahler manifolds.