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Summary:
Summary: There are two classical languages for analytic cohomology: Dolbeault and \v{C}ech. In some applications, however (for example, in describing the Penrose transform and certain representations), it is convenient to use some nontraditional languages. In [{\it M. G. Eastwood}, {\it S. G. Gindikin} and {\it 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.\par In this article we indicate a more general construction, which includes a version of \v{C}ech 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 \v{C}ech cohomology. Instead, it is possible to construct the !
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