Given a submersive morphism of complex manifolds f: X --> Y, and a complex vector bundle E on X, there is a relationship between the higher direct images of the sheaf of holomorphic sections of E and the index of the relative Dolbeault compplex twisted by E. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torus T 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 on T (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 papare may be applied to the study of the Mukai transform for more general varieties.
Fourier-Mukai transform and index theory
BARTOCCI, CLAUDIO;
1994-01-01
Abstract
Given a submersive morphism of complex manifolds f: X --> Y, and a complex vector bundle E on X, there is a relationship between the higher direct images of the sheaf of holomorphic sections of E and the index of the relative Dolbeault compplex twisted by E. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torus T 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 on T (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 papare may be applied to the study of the Mukai transform for more general varieties.
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