Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces

TL;DR

This paper derives explicit formulas for characteristic classes of tangent and tautological bundles on Hilbert schemes of points on non-compact simply-connected surfaces, advancing understanding of their geometric invariants.

Contribution

It provides the first closed formulas for multiplicative characteristic classes and Chern characters of tangent bundles on these Hilbert schemes, including tautological bundles.

Findings

Closed formula for multiplicative characteristic class of tangent bundle

Explicit expression for Chern character of tangent bundle

Formulas for characteristic classes of tautological bundles

Abstract

We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.