Tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence

TL;DR

This paper describes how tensor products of tautological bundles on Hilbert schemes of points relate under a specific derived equivalence, leading to new formulas for their cohomological invariants, including Euler characteristics.

Contribution

It provides a detailed description of the image of tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence, resulting in new cohomological formulas.

Findings

Formulas for Euler characteristics of tensor products on Hilbert schemes of two points.

Explicit formulas for triple tensor products.

Enhanced understanding of cohomological invariants of tautological bundles.

Abstract

We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.