Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic

TL;DR

This paper proves that Hodge isometries between certain hyper-Kahler varieties of K3[n]-type are algebraic, establishing a link between geometric and cohomological symmetries in these complex manifolds.

Contribution

It demonstrates that Hodge isometries of second cohomology are induced by algebraic correspondences and extends this to total cohomology with Mukai pairings.

Findings

Hodge isometries are induced by analytic correspondences.

Lifts to total cohomology preserve Mukai pairings.

Correspondences are algebraic in the projective case.

Abstract

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of their second rational cohomologies with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic.