Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)

TL;DR

This paper investigates the boundary behavior and singularities of an invariant associated with certain hyperkähler manifolds with involution, revealing algebraic properties and connections to modular forms.

Contribution

It demonstrates the algebraicity of the invariant's singularity and links it to Yoshikawa's invariant and modular forms for specific cases.

Findings

Proves algebraicity of the invariant's singularity.

Shows the invariant coincides with Yoshikawa's invariant in some cases.

Expresses the invariant as a Petersson norm of a Borcherds product and a Siegel modular form.

Abstract

We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.