K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces

TL;DR

This paper investigates the K-theoretic descendent series for Hilbert schemes of points on surfaces, proving their rationality by controlling equivariant holomorphic Euler characteristics and modifying a Macdonald polynomial identity.

Contribution

It introduces a new approach to establish the rationality of K-theoretic descendent series using a modified Macdonald polynomial identity.

Findings

Proves the rationality of K-theoretic descendent series.

Controls equivariant holomorphic Euler characteristics on Hilbert schemes.

Modifies a Macdonald polynomial identity for the analysis.

Abstract

We study the holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit.