# Higher rank Segre integrals over the Hilbert scheme of points

### Alina Marian

Northeastern University, Boston, USA### Dragos Oprea

University of California San Diego, La Jolla, USA### Rahul Pandharipande

ETH Zürich, Switzerland

## Abstract

Let $S$ be a nonsingular projective surface. Each vector bundle $V$ on $S$ of rank $s$ induces a tautological vector bundle over the Hilbert scheme of $n$ points of $S$. When $s=1$, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of the tautological bundles for all ranks $s$ over all $K$-trivial surfaces. Furthermore, in rank $s=2$, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.

## Cite this article

Alina Marian, Dragos Oprea, Rahul Pandharipande, Higher rank Segre integrals over the Hilbert scheme of points. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2979–3015

DOI 10.4171/JEMS/1149