Intersection forms of toric hyperkaehler varieties

Abstract

This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L^2 harmonic forms on complete hyperkaehler manifolds of linear growth.