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Nonclassical descriptions of analytic cohomology. (English) Zbl 1028.32004
Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12-19, 2002. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 67-72 (2003).
Summary: There are two classical languages for analytic cohomology: Dolbeault and Čech. In some applications, however (for example, in describing the Penrose transform and certain representations), it is convenient to use some nontraditional languages. In [M. G. Eastwood, S. G. Gindikin and H.-W. Wong, J. Geom. Phys. 17, 231-244 (1995; Zbl 0861.22009)] was developed a language that allows one to render analytic cohomology in a purely holomorphic fashion.
In this article we indicate a more general construction, which includes a version of Čech cohomology based on a smoothly parametrized Stein cover. The idea of this language is that, usually, there are only infinite Stein coverings of the complex manifold in question but, often, we can find natural Stein coverings parametrized by an auxiliary smooth manifold. Under these circumstances, it is unnatural to work with classical Čech cohomology. Instead, it is possible to construct the analytic cohomology from the de Rham complex on the parameter space but with holomorphic dependence in the corresponding Stein subset. This switch of language is rather like replacing sums by integrals to pass from discrete to continuous.
This material was the subject of a lecture presented by one of us (MGE) at the 22nd Czech Winter School on Geometry and Physics held in Srní in January 2002. This article contains only an outline and a couple of examples. Precise proofs will appear elsewhere.
For the entire collection see [Zbl 1014.00011].

32C35 Analytic sheaves and cohomology groups