Lefschetz decompositions of Kudla-Millson theta functions

2024/06/28 by Jan Hendrik Bruinier, Riccardo Zuffetti, Bruinier, Jan Hendrik +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Mathematics #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.2406.19921

openalex publication_date 2024/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the 1980'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 hyperkähler manifolds.

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• Extremal divisors on moduli spaces of K3 surfaces

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