Hodge classes of type (2, 2) on Hilbert squares of projective K3 surfaces

TL;DR

This paper constructs explicit bases for rational and integral Hodge classes of type (2,2) on Hilbert squares of general projective K3 surfaces, combining algebraic and lattice-theoretic methods.

Contribution

It provides the first explicit bases for these Hodge classes using Nakajima operators, algebraic models, and lattice theory, advancing understanding of the cohomology of Hilbert squares.

Findings

Explicit basis for rational Hodge classes of type (2,2)

Explicit basis for integral Hodge classes of type (2,2)

Application of Nakajima operators and lattice theory

Abstract

We give a basis for the vector space generated by rational Hodge classes of type (2,2) on the Hilbert square of a projective K3 surface general in its rank, which is a subspace of the singular cohomology ring with rational coefficients: we use Nakajima operators and an algebraic model developed by Lehn and Sorger as main tools. We then obtain a basis of the lattice generated by integral Hodge classes of type (2,2) on the Hilbert square of a projective K3 surface general in its rank: we exploit lattice theory, a theorem by Qin and Wang and a result by Ellingsrud, G\"ottsche and Lehn.