  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.
