A type of comparison theorem in polynomial growth cohomology
This note is a continuation of our previous paper on stratification theory and cohomology with some structures (growth and division), which was applied to give a generalization of the algebraic and analytic de Rham theorems of A. Grothendieck [the author, ''Cohomology with polynomial growth and completion theory'', Publ. Res. Inst. Math. Sci. 17, 371-552 (1981)]. In the present note, we point out that a type of a GAGA comparison is given in terms of polynomial growth cohomology and a stratification theory. This may be a generalization of the original GAGA comparison of \textit{J.-P. Serre} [Ann. Inst. Fourier 6(1955/56), 1-42 (1956; Zbl 0075.304)] (for a projective variety) and similar comparisons for locally free sheaves over a quasi projective variety [\textit{M. Cornalba} and \textit{D. Griffiths}, Invent. Math. 28, 1-106 (1975; Zbl 0293.32026)] and \textit{G. Maltsiniotis} (after \textit{P. Deligne}) [Sémin. Géométrie analytique Éc. Norm. Super. 1972-1973, Astérisque 17, 141-160 (1974; Zbl 0297.14006)]. The fundamental importance of the stratification theory for analytic varieties has been well known and is increasing. The stratification theoretical cohomology group \(H_{str}\) (in the paper under review) may be useful for some global treatments of coherent sheaves and other analytic objects on analytic varieties. As I mentioned in the present paper, the stratification theoretical method works well for some explicit constructions of holomorphic bundles [''A stratification theoretical method of construction of holomorphic vector bundles'', submitted to Proc. U.S.-Japan Semin. Singularities, held at July, 1984].
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