Cohomology of singular hypersurfaces
type: Item
instance of: scholarly article
title: Cohomology of singular hypersurfaces
zbMATH Open document ID: 1024.11046
author: Bernard M. Dwork
published in: Pacific Journal of Mathematics
publication date: 2000-12-03T00:00:00Z
review text: Professor Dwork passed away on May 9, 1998. This paper was completed only a few days earlier. This article is tightly related to previous work of the author on the zeta function of singular hypersurfaces [\textit{B. Dwork}, Ann. Math. (2) 83, 457--519 (1966; Zbl 0173.48602)], and to the Padova thesis of \textit{C. Bertolin} [``\(G\)-fonctions et cohomologie des hypersurfaces singulières'', Bull. Aust. Math. Soc. 55, 353--383 (1997; Zbl 0927.11038)]. The author is interested in a variation of his relative cohomology groups in families of singular hypersurfaces. He reproves the main result of Bertolin (loc. cit.), namely that the Picard-Fuchs equations are of arithmetic (``\(G\)'') type in a simpler way, and extends to this relative situation some results of his own (loc. cit.). Dwork's cohomology of hypersurfaces consists of a direct and of a dual theory, in both the algebraic and the \(p\)-adic analytic settings. In the case of non-singular hypersurfaces, these cohomology spaces have been interpreted in terms of de Rham or rigid cohomology for the direct theory and as de Rham or rigid cohomology with ``compact supports'' for the dual theory. So, in the case of a (generically) non-singular family, the result of this paper says that \(G\)-coefficients are stable by direct image. This has been proven in a more standard way in \textit{Y. Andre} and \textit{F. Baldassarri} [``Geometric theory of \(G\)-functions'', in: F. Catanese (ed.), Arithmetic geometry, Cambridge Univ. Press, Symp. Math. 37, 1--22 (1997; Zbl 0936.14014)]. But the interpretation of Dwork's cohomology spaces for singular hypersurfaces is yet unknown. It is impressive that the variation of these spaces should still obey the same arithmetic constraint.
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 1537855
zbMATH Keywords: relative cohomology groups
zbMATH Keywords: families of singular hypersurfaces
zbMATH Keywords: Picard-Fuchs equations
reviewed by: Francesco Baldassarri
MaRDI profile type: Publication
OpenAlex ID: W2004366695
DOI: 10.2140/PJM.2000.195.81
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description: scientific article; zbMATH DE number 1537855