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A Grothendieck-Riemann-Roch formula for maps of complex manifolds. (English) Zbl 0539.14005
This is a detailed proof of the formula announced previously by the authors [Bull. Am. Math. Soc., New Ser. 5, 182-184 (1981; Zbl 0495.14010)]. This is a G.R.R. formula relating characteristic classes of a coherent sheaf \({\mathcal F}\) to those of its direct images under a map of complex manifolds f:\(X\to Y\), arbitrary except for the condition that f is proper on the support of \({\mathcal F}\). Except in the case where Y is a point, previous proofs of such formulae required that X and Y be subvarieties of projective spaces. By contrast, this proof uses local geometric formulae for Čech cochains representing the characteristic classes, derives the appropriate local relations between these cochains and leads to a G.R.R. formula relating classes in Hodge cohomology rather than ordinary cohomology.
As in the authors’ previous work on Hirzebruch-Riemann-Roch, the local formulae are based on the notion of a ’twisting cochain’. Innovations here include the introduction of twisting cochains for perfect complexes of sheaves, direct images of twisting cochains and the use of Grauert’s coherence theorem and the relative version of Serre-Grothendieck duality.

MSC:
14C40 Riemann-Roch theorems
32Q99 Complex manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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