Curvature and characteristic numbers of hyperk\"ahler manifolds

Abstract

We express characteristic numbers of compact hyperk\"ahler manifolds in graph-theoretical form, considering them as a special case of the curvature invariants introduced by Rozansky and Witten. The appropriate graphs are generated by ``wheels'' and we use the recently proved Wheeling Theorem to give a formula for the L2 norm of the curvature of an irreducible hyperk\"ahler manifold in terms of the volume and Pontryagin numbers. The formula involves the multiplicative sequence which is the square root of the A-hat polynomial.