Rigid ideal sheaves and modular forms

TL;DR

This paper derives an explicit formula for the generating series of Euler characteristics of G-invariant zero-dimensional subschemes on a surface with ADE singularities, showing it forms a holomorphic modular form.

Contribution

It provides a new explicit expression linking geometry of invariant subschemes to modular forms in the context of surfaces with ADE singularities.

Findings

Generating series is a holomorphic modular form.

Explicit formula for Euler characteristics of G-invariant subschemes.

Connection between surface geometry and modular forms.

Abstract

Let $X$ be a complex smooth quasi-projective surface acted upon by a finite group $G$ such that the quotient $X/G$ has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics of the zero-dimensional components in the moduli space of zero-dimensional subschemes on $X$ invariant under the action of $G$. We show that this generating series (up to a suitable rational power of the formal variable) is a holomorphic modular form.