Exact Formulas for Invariants of Hilbert Schemes

TL;DR

This paper derives exact formulas for invariants of Hilbert schemes of points on surfaces using the circle method, connecting modular forms with geometric invariants like signature and Euler characteristic.

Contribution

It introduces a novel application of the circle method to obtain convergent series for Hilbert scheme invariants, enhancing understanding of their structure and asymptotic behavior.

Findings

Exact formulas for signature and Euler characteristic of Hilbert schemes

Convergent series representations for these invariants

Analysis of asymptotic and distributional properties of coefficients

Abstract

A theorem of G\"ottsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in certain infinite product $q$-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these $q$-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the $q$-series' coefficients.