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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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