The combinatorics of Lehn's conjecture

TL;DR

This paper proves key vanishings in Lehn's conjecture using residue calculations, extends the analysis to symmetric powers of curves, and proposes a comprehensive conjecture for higher rank bundles on surfaces.

Contribution

It provides a proof of the vanishings needed for Lehn's conjecture, extends the framework to symmetric powers of curves, and formulates a new conjecture for higher rank bundles on surfaces.

Findings

Proved vanishings required by Voisin for Lehn's conjecture.

Extended the analysis to symmetric powers of curves with higher rank bundles.

Proposed a complete conjecture for top Segre classes on Hilbert schemes for higher rank bundles.

Abstract

Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result of this short note is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a smooth projective curve C associated to a higher rank vector bundle V on C. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of S associated to a higher rank vector bundle on S in the K-trivial case.