The canonical class and Chern character operators on Hilbert schemes of points

Abstract

Let $X$ be a smooth projective complex surface. We obtain explicit formulas for the Chern character operators on the Hilbert schemes of points $X^{[n]}$ in terms of Nakajima operators, determining in particular the coefficients of the terms involving the canonical class $K_X$ and its square. Consequently, the cohomology ring structure of the Hilbert scheme is determined. As an application, we give an alternative proof of the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the cohomological Fock space $\mathbb{H}_X = \bigoplus_{n\geq 0} H^*(X^{[n]},\mathbb{Q})$.