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Rationality problems and conjectures of Milnor and Bloch-Kato. (English) Zbl 1279.14063
The classical Lüroth problem may be formulated as the question whether a unirational variety $$X/k$$ is in fact rational. It is known that in general the answer is negative. For $$k={\mathbb C}$$ the field of complex numbers M. Artin and D. Mumford [Proc. Lond. Math. Soc., III. Ser. 25, 75–95 (1972; Zbl 0244.14017)] following a suggestion of C. P. Ramanujam showed that for a smooth complex variety $$X$$ the torsion subgroup of the singular cohomology group $$H^{3}(X,{\mathbb Z})$$ is a birational invariant. By constructing a conic bundle over a rational surface and exhibiting a $$2$$-torsion class in the above group they found an example of a unirational variety that is not rational.
In the paper under review, the author uses unramified cohomology groups to detect counterexamples to the Lüroth problem. The unramified cohomology groups (for any $$i$$ and $$j$$) were defined By J.-L. Colliot-Thélène and M. Ojanguren [Invent. Math. 97, No. 1, 141–158 (1989; Zbl 0686.14050)] as the subgroups $$H^{i}_{ur}(L/k,{{\mu}_{n}^{{\otimes}j}})$$ of $$H^{i}_{\text{ét}}(L,{{\mu}_{n}^{{\otimes}j}})$$ consisting of the unramified elements at every discrete valuation of $$L$$ trivial on $$k.$$ Recall that a class $${\alpha}\in H^{i}_{\text{ét}}(L,{{\mu}_{n}^{{\otimes}j}})$$ is unramified at a discrete valuation $$\nu$$ of $$L/k$$ if $$\alpha$$ is in the image of the restriction map $$H^{i}_{\text{ét}}(A,{{\mu}_{n}^{{\otimes}j}})\rightarrow H^{i}_{\text{ét}}(L,{{\mu}_{n}^{{\otimes}j}})$$ , where $$A$$ is the valuation ring associated with $$\nu$$. The author generalizes the method of E. Peyre [Math. Ann. 296, No. 2, 247–268 (1993; Zbl 0790.12001)] and constructs for any prime $$l$$ and $$n\geq 2$$ a rationally connected, non-rational variety. The non-rationality of this variety is detected by a non-trivial class of degree $$n$$ in its unramified cohomology. By definition for a variety $$X$$, $$H^{i}_{ur}(X,{{\mu}_{n}^{{\otimes}j}}):= H^{i}_{ur}(k(X)/k,{{\mu}_{n}^{{\otimes}j}})$$. For $$l=2$$ these varieties are unirational and their non-rationality cannot be detected by a torsion unramified class of lower degree. The techniques used in the paper follow (to some extent) those used in Voevodsky’s proof of the Milnor conjecture and the Voevodsky-Rost proof of the Bloch-Kato conjecture.

##### MSC:
 14M20 Rational and unirational varieties 19D45 Higher symbols, Milnor $$K$$-theory 11E81 Algebraic theory of quadratic forms; Witt groups and rings 12G05 Galois cohomology 14F42 Motivic cohomology; motivic homotopy theory
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