Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

2022/01/23 by Sailun Zhan, Zhan, Sailun
Mathematics · #14G17 #14J15 #14J50 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2201.09215

openalex publication_date 2022/01/23 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

Göttsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface S admits an action by a finite group G, we describe the action of G on the Hodge pieces via point counting. Each element of G gives a trace on ∑_n=0^∞∑_i=0^∞(-1)ⁱHⁱ(S^[n],ℂ)qⁿ. In the case that S is a K3 surface or an abelian surface, the resulting generating functions give some interesting modular forms when G acts faithfully and symplectically on S.

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