Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant

TL;DR

This paper introduces an invariant for certain holomorphic symplectic fourfolds with involution, using equivariant analytic torsion, and provides a formula for its complex Hessian, advancing understanding of their geometric properties.

Contribution

It constructs a new invariant for $K3^{[2]}$-type manifolds with antisymplectic involution using equivariant analytic torsion and derives a formula for its complex Hessian.

Findings

Constructed an invariant for holomorphic symplectic fourfolds with involution.

Derived a formula for the complex Hessian of the invariant.

Enhanced understanding of the geometric structure of these manifolds.

Abstract

In this paper, we construct an invariant for irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type with antisymplectic involution by using the equivariant analytic torsion. Moreover, we give a formula for the complex Hessian of the logarithm of the invariant.