Intersection forms of toric hyperkähler varieties

Abstract

This note proves combinatorially that the intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkähler variety is always definite, providing a large number of non-trivial $L^2$ harmonic forms for toric hyperkähler 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 hyperkähler manifolds of linear growth.