On a conjucture of Kashiwara relating Chern and Euler classes of \(\mathcal{O}\)-modules. (English) Zbl 1247.32013

Characteristic classes in Hochschild homology are well known in homological algebra and have been recently studied in various algebraico-geometric contexts. Recall that M. Kashiwara had constructed, for every object \(\mathcal{F}\) of the bounded derived category of analytic sheaves on a complex manifold \(X\) with coherent cohomology, the Hochschild and co-Hochschild classes of \(\mathcal{F}\) which are mapped via Hochschild-Kostant-Rosenberg isomorphisms to the so-called Chern and Euler classes of \(\mathcal{F}\). The aim of this paper is to give a simple proof of Kashiwara’s conjecture that states that the Euler class of a coherent analytic sheaf \(\mathcal{F}\) on a complex manifold \(X\) is the product of the Chern character of \(\mathcal{F}\) with the Todd class of \(X\). As a corollary a simpler and functorial proof of the Grothendieck-Riemman-Roch theorem in Hodge cohomology for complex manifolds is obtained.
The high interest of the proofs presented in this paper is complemented with a clear and objective writing.


32C35 Analytic sheaves and cohomology groups
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