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Lefschetz decompositions of Kudla-Millson theta functions

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arxiv 2406.19921 v2 pith:N4JZMC6I submitted 2024-06-28 math.NT math.AG

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keywords cohomologythetadecompositionformfunctionkudla-millsonlefschetzmodular
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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 hyperk\"ahler manifolds.

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  1. The Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces and generalizations

    math.AG 2024-11 conditional novelty 8.0 of 10

    The Picard group of the Baily-Borel compactification of the moduli space of quasi-polarized K3 surfaces is Z, spanned by the extended Hodge line bundle.

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