Fourier-Mukai transform and index theory

Given a submersive morphism of complex manifoldsf: X→Y, and a complex vector bundleE onX, there is a relationship between the higher direct images of ε (the sheaf of holomorphic sections ofE) and the index of the relative Dolbeault complex twisted byE. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torusT of complex dimension 2 and the Nahm transform of instantons. We also offer a proof of Mukai’s inversion theorem which circumvents the use of derived categories by introducing spectral sequences of sheaves onT (this is related to Donaldson and Kronheimer’s proof, but is automatically global and somehow simpler). The general framework developed in the first part of this paper may be applied to the study of the Mukai transform for more general varieties. Research partly supported by the Italian Ministry for University and Research through the research projects ‘Metodi geometrici e probabilistici in fisica matematica’ and ‘Geometria reale e complessa,’ and by the Spanish DGICYT through the research projects PB91-0188 and PB92-0308. This paper has been typeset by using theAMS-TEX macro package