K3 Surfaces with Involution and Analytic Torsion

1998/08/31 by Kenichi Yoshikawa, Ken-Ichi Yoshikawa, Yoshikawa, Ken-Ichi
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG

paper · pdf · doi:10.48550/arxiv.math/9808129

AMS-Tex, no figure, new title

openalex publication_date 1998/08/31 · arxiv created 1999/07/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this paper), and show that it (together with the analytic torsion of the fixed curves) can be identified with the automorphic form on the moduli space characterizing the discriminant locus. Three lattices A₁, II_1,1(2), II_1,9(2) are of particular interest, because they consist of the building blocks of 2-elementary lattices. An explicit formula is given for them. In particular, if M is twice the Enriques lattice, the automorphic form coincides with Borcherds's Phi-function which confirms an observation by Jorgenson-Todorov and Harvey-Moore. Some other examples are shown to be related to Borcherds's product and generalized Kac-Moody algebras.

Citations

• On the parabolic kernel of the Schrödinger operator

Related

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• K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant
• Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)
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