On the De Rham cohomology of algebraic varieties. (English) Zbl 0145.17602

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[1] M. F. Attyah andW. V. D. Hodge, Integrals of the second kind on an algebraic variety,Annals of Mathematics, vol. 62 (1955), p. 56–91. This paper is referred to by A-H in the sequel. · Zbl 0068.34401 · doi:10.2307/2007100
[2] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero,Annals of Maths., vol. 79 (1964), p. 109–326. · Zbl 0122.38603 · doi:10.2307/1970486
[3] J.-P. Serre, Géométric algébrique et géométrie analytique,Annales de l’Institut Fourier, vol. VI (1956), p. 1–42.
[4] H. Grauert etR. Remmert, Faisceaux analytiues cohérents sur le produit d’un espace analytique et d’un espace projectif,C.R. Acad. Sc. Paris, t. 245, p. 819–822. The proof that follows is essentially the same as the one given in the previous remarks (6), except that reference to GAGA is replaced by a reference to the theorem of Grauert-Remmert, which should be viewed as the generalization of GAGA, from the case of a base space reduced to one point, to the case of a ground space an arbitrary complex analytic space. For the general philosophy of the result of Grauert and Remmert, one may read my talk in Cartan’s Seminar 1960–61, Exposé 15, Rapport sur les théorèmes de finitude de Grauert et Remmert (especially the remarks on the last page of the exposé). We sould remarks also that the theorem of Grauert and Remmert is implicit in the proof of the isomorphism \(g_* (\mathcal{K}' \bullet ) \simeq \mathcal{K} \bullet \) (which in our remark (6) above corresponds to the isomorphism (8”)).
[5] Cf.M. Artin,Grothendieck Topologies, Spring, 1962, Harvard University, orM. Artin etA. Grothendieck,Cohomologie étale des Schémas, Séminaire de Géométrie algébrique de L’I.H.E.S., 1963–64.
[6] This ”hope”, and the next one, are excessive, as Serre pointed out. Indeed, it is not hard to check that in his example of a non singular surface X, quotient of a regular surface inP 3 by the group G =Z/p Z operating freely,p = characteristic (cf.J.-P. Serre, Sur la topologie des variétés algébriques en caractéristiquep, Symposium Internacional de Topologia Algebrica (1958), p. 24–53, proposition 16), one has dim H1(X)=i, whereas {\(\pi\)}1(X)=Z/p Z and hence the first Betti numberb 1(X) is zero. Thus the dimensions of H i (X) may still be too big, probably due to torsion phenomena in thep-adic cohomology.
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