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The Riemann-Roch theorem for complex spaces. (English) Zbl 0627.32004
A proof of Riemann-Roch theorem in the following form is given: Denote by \(K_ 0^{hol}(M)\) the Grothendieck group of the category of all coherent sheaves on the complex space M, and by \(K_ 0^{top}(M)\) the usual homology K-functor of the underlying topological space. There exists a group morphism \(\alpha_ M: K_ 0^{hol}(M) \to K_ 0^{top}(M)\) such that: (a) For M regular the restriction of \(\alpha_ M\) to the subgroup \(K^ 0_{hol}\) generated by the classes of all locally free sheaves coincides with the natural morphism to \(K^ 0_{top}(M)\simeq K_ 0^{top}(M)\) attaching to a locally free sheaf the class of the corresponding vector bundle. (b) If \(f: M\to N\) is a proper morphism of complex spaces, and \(f_ !:K_ 0^{hol}(M)\to K_ 0^{hol}(N)\) is the direct image homomorphism provided by Grauert’s theorem, then the equality \(f_*\alpha_ M({\mathcal L})=\alpha_ N(F_ !{\mathcal L})\) holds for any coherent sheaf \({\mathcal L}\) on M. When U is open in \({\mathbb{C}}^ n\), and \({\mathcal B}\) is any Stein covering of U, \(\alpha_ U({\mathcal L})\) is represented by the parametrized Koszul complex \(KC_{\bullet}(U,{\mathcal B},{\mathcal L})(\lambda)\) associated to the coordinates \((z_ 1,...,z_ n)\) of \({\mathbb{C}}^ n\) acting as multiplication endomorphisms in the standard cochain complex \(C_{\bullet}(U,{\mathcal B},{\mathcal L})\) of \({\mathcal B}\). To deal with the singularities of M, the globalization is achieved using a so-called almost holomorphic embedding of M in \({\mathbb{R}}^{2N}\). Rather than \(\alpha_ M({\mathcal L})\) itself, its Alexander dual in \(K^ 0({\mathbb{R}}^{2N},{\mathbb{R}}^{2N}/M)\) is then constructed.
A new proof of Grauert’s direct image theorem has been given by the author [Proc. Am. Math. Soc. 99, 535-542 (1987; Zbl 0625.46044)], which is derived from a parametrized version of L. Schwartz theorem on compact perturbations of epimorphism used in the present proof.
Reviewer: F.Campana

MSC:
32C35 Analytic sheaves and cohomology groups
55N15 Topological \(K\)-theory
14C40 Riemann-Roch theorems
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R22 Topology of vector bundles and fiber bundles
57R20 Characteristic classes and numbers in differential topology
32C15 Complex spaces
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