VANISHING THEOREMS FOR QUATERNIONIC COMPLEXES

Abstract

1. Introduction

In 1978, Atiyah, Hitchin and Singer [1] introduced the twistor space as a ℙ¹-fibration over a half-conformally flat 4-manifold, and thereby established a beautiful correspondence between Yang–Mills fields on 4-manifolds and holomorphic vector bundles on complex 3-folds. Hitchin gave the Penrose correspondence between solutions of anti-self-dual zero rest-mass field equations on a half-conformally flat manifold M and the sheaf cohomology groups H¹(Z, F(k)), k[les ]−2, on its twistor space Z [4]. Here the holomorphic bundle F is the pull-back bundle of an anti-self-dual bundle over M, and we denote by [Oscr ](−1) the tautological line bundle associated to the half-spin bundle. Hitchin further showed how H¹(Z, F(−1)) corresponds to solutions of the self-dual Dirac equations, and interpreted H¹(Z, F(k)), k[ges ]0, in terms of the cohomology of an elliptic complex on M.