Colliot-Thélène, Jean-Louis; Pirutka, Alena Third unramified cohomology group of a cubic threefold over a function field in one variable. (Troisième groupe de cohomologie non ramifiée d’un solide cubique sur un corps de fonctions d’une variable.) (French. English summary) Zbl 1419.14012 Épijournal de Géom. Algébr., EPIGA 2, Article No. 13, 13 p. (2018). The main result of the paper (Theorem 1.1) is the proof of the integral Hodge conjecture in codimension two for complex varieties of dimension four admitting a fibration over a smooth curve whose generic fiber is a smooth cubic of dimension three. A weak version of this result was proved by Voisin assuming that the fibration has only mild singularities [C. Voisin, J. Algebr. Geom. 22, No. 1, 141–174 (2013; Zbl 1259.14006)].Some reduction steps (using unramified cohomology) allow to deduce the main result from the following fact (Theorem 1.2). Let \(k\) be the function field of a complex curve, \(G\) be its absolute Galois group and \(X\) be a smooth cubic of dimension three. Then the canonical map \(\mathrm{CH}^2(X) \longrightarrow \mathrm{CH}^2(X_{\bar{k}})^G\) is surjective.For such a special variety \(X\) these Chow groups have an explicit geometric description and the surjectivity of the map is cleverly reduced to the celebrated theorem of Graber, Harris and Starr [T. Graber et al., J. Am. Math. Soc. 16, No. 1, 57–67 (2003; Zbl 1092.14063)].These geometric descriptions of the Chow groups of cubics go back to Murre, Iliev, Markushevich and Tikhomirov. The authors need to extend these previous works. Such a foundational material is of independent interest. Reviewer: Giuseppe Ancona (Strasbourg) Cited in 1 Document MSC: 14C25 Algebraic cycles 14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry 14D06 Fibrations, degenerations in algebraic geometry 14M22 Rationally connected varieties 14H40 Jacobians, Prym varieties 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) Keywords:Hodge conjecture; cubics; unramified cohomology; rationally connected varieties; intermediate Jacobians Citations:Zbl 1259.14006; Zbl 1092.14063 PDFBibTeX XMLCite \textit{J.-L. Colliot-Thélène} and \textit{A. Pirutka}, Épijournal de Géom. Algébr., EPIGA 2, Article No. 13, 13 p. (2018; Zbl 1419.14012) Full Text: arXiv Link