Washnitzer's conjecture and the cohomology of a variety with a single isolated singularity (Q1059684)

Let X be an irreducible complex quasi-projective variety. There exists a natural continuous map \(\pi\) : \(X_{Class}\to X_{Zar}\) which yields the Leray spectral sequence \(H^ p_{Zar}(X,R^ q\pi_*({\mathbb{C}}))\) converging to \(H^{p+q}(X,{\mathbb{C}})\) (here \(H^*(X,{\mathbb{C}})\) denotes the classical singular cohomology). Bloch and Ogus proved the Washnitzer conjecture to the effect that, for a nonsingular X, the decreasing filtration in \(H^{p+q}(X,{\mathbb{C}})\) induced by this spectral sequence coincides with the ''coniveau'' filtration \(N^ pH^ m=\cup Ker\{H^ m(X,{\mathbb{C}})\to H^ m(X\setminus Z,{\mathbb{C}})\},\) where Z runs through the set of Zariski closed subvarieties of X of codimension at least p [cf. \textit{S. Bloch} and \textit{A. Ogus}, Ann. Sci. Éc. Norm. Super., IV. Sér. 7(1974), 181-201 (1975; Zbl 0307.14008)]. In the paper under review the author extends this result to the case of varieties with at most one isolated singularity (the author calls them ''almost nonsingular'') the coniveau filtration being defined as above, where Z runs through the set of Zariski closed subvarieties which do not contain the singular point. Moreover, the Leray spectral sequence coincides with the spectral sequence of hypercohomology associated to the de Rham complex [cf. \textit{R. Hartshorne}, Publ. Math., Inst. Haut. Étud. Sci. 45, 5-99 (1976; Zbl 0326.14004)].