Higher rank Segre integrals over the Hilbert scheme of points

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.