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

TL;DR

This paper fully characterizes a holomorphic torsion invariant of K3 surfaces with involution as an automorphic function, expressing it via Borcherds lifts and Siegel modular forms, and explores its twisted version and related conjecture.

Contribution

It provides a complete description of the torsion invariant's structure as an automorphic function, including its modularity and the twisted version, advancing understanding of K3 surface invariants.

Findings

Invariant expressed as product of Borcherds lift and Siegel modular form

Proved modularity and uniqueness of the twisted invariant

Studied an equivariant analogue of Borcherds' conjecture

Abstract

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. 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.