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Unramified cohomology and the entire Hodge conjecture. (Cohomologie non ramifiée et conjecture de Hodge entière.) (French. English summary) Zbl 1244.14010
Let \(X\) be a complex projective manifold of dimension \(d\). For \(i \geq 1\) we set \(\mathbb Z(i):=\mathbb Z(1)^{\otimes i}\) where \(\mathbb Z(1):=\mathbb Z(2 \pi i) \subset \mathbb C\). We define the group of integral Hodge classes \[ Hdg^{2i}(X, \mathbb Z) \subset H^{2i}(X(\mathbb C), \mathbb Z(i)) \] as the preimage of the usual Hodge classes \(Hdg^{2i}(X, \mathbb Q)\) under the natural map from integral to rational cohomology. We define \(H^{2i}_{ alg}(X(\mathbb C), \mathbb Z(i))\) as the image of the map from the Chow group \[ CH^i(X) \rightarrow H^{2i}(X(\mathbb C), \mathbb Z(i)). \] Then we have an inclusion \(H^{2i}_{ alg}(X(\mathbb C), \mathbb Z(i)) \subset Hdg^{2i}(X, \mathbb Z)\) and we set \[ Z^{2i}(X) := Hdg^{2i}(X, \mathbb Z)/H^{2i}_{ alg}(X(\mathbb C), \mathbb Z(i)). \] If the Hodge conjecture holds in degree \(2i\) the group \(Z^{2i}(X)\) is torsion and equal to \[ H^{2i}(X(\mathbb C), \mathbb Z(i)) / H^{2i}_{ alg}(X(\mathbb C), \mathbb Z(i)), \] in particular it would be a finite group. If the group \(Z^{2i}(X)\) is equal to zero this means that the Hodge conjecture holds even for integral coefficients, a statement that is true for \(i=1\), but known to be false in general for \(i=2\) by examples of M. F. Atiyah and F. Hirzebruch [Topology 1, 25–45 (1962; Zbl 0108.36401)] and J. Kollár [in: Classification of irregular varieties. Minimal models and Abelian varieties. Lecture Notes in Mathematics. 1515. Berlin etc.: Springer-Verlag (1992; Zbl 0744.00029), Lemma on p. 134].
In the paper under review the authors consider the group \(Z^4(X)\) which is particularly interesting since it is a birational invariant of the manifold \(X\). Their main theorem relates this group to unramified cohomology groups \(H^i_{ nr}(X, \mathbb Z/n)\) introduced by J.-L. Colliot-Thélène and M. Ojanguren [Invent. Math. 97, No. 1, 141–158 (1989; Zbl 0686.14050)]. More precisely they prove the following statement: let \(X\) be a complex projective manifold such that the Chow group of zero cycles \(CH_0(X)\) has support on a surface. Then we have an isomorphism of finite groups \[ H^3_{ nr}(X, \mathbb Q/\mathbb Z(2)) \simeq Z^4(X) \] where the first group is the union of its subgroups \(H^3_{ nr}(X, \mu_n^{\otimes 2})\). Based on this fundamental result it is possible to compare and translate results obtained by methods of algebraic geometry and algebraic \(K\)-theory. For example it implies that for any smooth projective threefold that is covered by rational curves, the group \(H^3_{ nr}(X, \mathbb Q/\mathbb Z(2))\) is zero since \(Z^4(X)\) is zero by a theorem of C. Voisin [Advanced Studies in Pure Mathematics 45, 43–73 (2006; Zbl 1118.14011)]. Vice versa an example of a non-rational manifold by Colliot-Thélène and Ojanguren [loc. cit.] gives an example of a projective manifold of dimension \(6\) that is unirational, but \(Z^4(X)=Z^4(X)_{ tors} \neq 0\). This gives a negative answer to a question of C. Voisin [Jpn. J. Math. (3) 2, No. 2, 261–296 (2007; Zbl 1159.14005)]. The authors also consider the case of a threefold admitting a fibration \(X \rightarrow \Gamma\) onto a smooth curve, under certain conditions they establish a link between the group \(H^3_{ nr}(X, \mathbb Q/\mathbb Z(2))\), the group \(Z^4(X)\) and the index of the fibration, i.e., the greatest common divisor of the degrees of all the multisections.

MSC:
14E05 Rational and birational maps
14F99 (Co)homology theory in algebraic geometry
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