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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

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.

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)
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