The cup product of the Hilbert scheme for K3 surfaces

TL;DR

This paper constructs a sequence of Frobenius algebras associated with a given algebra and establishes a canonical isomorphism with the cohomology of Hilbert schemes of K3 surfaces, revealing deep algebraic structures.

Contribution

It introduces a new algebraic framework linking Frobenius algebras to the cohomology of Hilbert schemes of K3 surfaces, providing a novel perspective on their structure.

Findings

Established a canonical ring isomorphism for Hilbert schemes of K3 surfaces.

Linked Frobenius algebra structures to geometric cohomology.

Provided a new algebraic approach to studying Hilbert schemes.

Abstract

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A^[n] in such a way that for any smooth projective surface X with trivial canonical divisor there is a canonical isomorphism of rings between (H*X)^[n] and the cohomology H*(X^[n]) of the n-th Hilbert scheme of X.