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K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: The structure of the invariant

Part of: Curves Discontinuous groups and automorphic forms Partial differential equations on manifolds; differential operators Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  19 November 2020

Shouhei Ma  and
Ken-Ichi Yoshikawa
Shouhei Ma
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Tokyo152-8551, Japanma@math.titech.ac.jp
Ken-Ichi Yoshikawa
Affiliation:
Faculty of Science, Department of Mathematics, Kyoto University, Kyoto606-8502, Japanyosikawa@math.kyoto-u.ac.jp
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Abstract

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

Keywords

K3 surfacesanalytic torsionBorcherds productstheta constants

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables) 14H45: Special curves and curves of low genus 14J28: $K3$ surfaces and Enriques surfaces 58J52: Determinants and determinant bundles, analytic torsion
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 10 , October 2020 , pp. 1965 - 2019
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Copyright
© The Author(s) 2020

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