Cohomology of singular hypersurfaces (Q1587571)

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.