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Rationality problems and conjectures of Milnor and Bloch–Kato

Part of: (Co)homology theory Forms and linear algebraic groups Higher algebraic $K$-theory Homological methods (field theory) Special varieties

Published online by Cambridge University Press:  03 June 2013

Aravind Asok
Aravind Asok*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA email [email protected]
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Abstract

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We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

Keywords

rationalityunramified cohomologyMilnor conjecture𝔸1-connectedness

MSC classification

Secondary: 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
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1312 - 1326
Copyright
© The Author(s) 2013 

References

Arason, J. Kr., Cohomologische invarianten quadratischer Formen, J. Algebra 36 (1975), 448–491.Google Scholar
Artin, M. and Mumford, D., Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3) 25 (1972), 75–95.Google Scholar
Asok, A. and Morel, F., Smooth varieties up to ${ \mathbb{A} }^{1} $-homotopy and algebraic $h$-cobordisms, Adv. Math. 227 (2011), 1990–2058.Google Scholar
Asok, A., Birational invariants and ${ \mathbb{A} }^{1} $-connectedness J. Reine. Angew. Math., doi:10.1515/crelle-2012-0034, April 2012.Google Scholar
Colliot-Thélène, J.-L., Birational invariants, purity and the Gersten conjecture, in K-theory and Algebraic Geometry: Connections with Quadratic forms and Division Algebras, Santa Barbara, CA, 1992, Proceedings of Symposia in Pure Mathematics, vol. 58 (American Mathematical Society, Providence, RI, 1995), 1–64.Google Scholar
Colliot-Thélène, J.-L., Hoobler, R. T. and Kahn, B., The Bloch–Ogus–Gabber theorem, in Algebraic K-theory, Toronto, ON, 1996, Fields Institute Communications, vol. 16 (American Mathematical Society, Providence, RI, 1997), 31–94.Google Scholar
Colliot-Thélène, J.-L. and Ojanguren, M., Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), 141–158.Google Scholar
Colliot-Thélène, J.-L. and Voisin, C., Cohomologie non Ramifiée et conjecture de Hodge entière, Duke Math. J. 161 (2012), 735–801.Google Scholar
Elman, R., Karpenko, N. and Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, vol. 56 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57–67.CrossRefGoogle Scholar
Grothendieck, A., Le groupe de Brauer. III. Exemples et compléments, in Dix Exposés sur la Cohomologie des Schémas (North-Holland, Amsterdam, 1968), 88–188.Google Scholar
Haesemeyer, C. and Weibel, C., Norm varieties and the chain lemma (after Markus Rost), in Algebraic Topology, Abel Symposia, vol. 4 (Springer, Berlin, 2009), 95–130.Google Scholar
Iskovskikh, V. A. and Prokhorov, Yu. G., Fano varieties, in Algebraic Geometry, V, Encyclopaedia of Mathematical Sciences, vol. 47 (Springer, Berlin, 1999), 1–247.Google Scholar
Izhboldin, O. T., Fields of $u$-invariant 9, Ann. of Math. (2) 154 (2001), 529–587.CrossRefGoogle Scholar
Jacob, B. and Rost, M., Degree four cohomological invariants for quadratic forms, Invent. Math. 96 (1989), 551–570.Google Scholar
Kahn, B., Motivic cohomology of smooth geometrically cellular varieties, in Algebraic K-theory, Seattle, WA, 1997, Proceedings of Symposia in Pure Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1999), 149–174.Google Scholar
Karpenko, N. and Merkurjev, A., On standard Norm varieties, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 175–214.Google Scholar
Kollár, J., Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32 (Springer, Berlin, 1996).Google Scholar
Kahn, B. and Sujatha, R., Motivic cohomology and unramified cohomology of quadrics, J. Eur. Math. Soc. (JEMS) 2 (2000), 145–177.CrossRefGoogle Scholar
Manin, Yu. I., Cubic Forms, North-Holland Mathematical Library, vol. 4, second edition (North-Holland Publishing Co., Amsterdam, 1986), translated from the Russian by M. Hazewinkel.Google Scholar
Mazza, C., Voevodsky, V. and Weibel, C., Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Providence, RI, 2006).Google Scholar
Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.Google Scholar
Merkurjev, A. and Suslin, A., Motivic cohomology of the simplicial motive of a Rost variety, J. Pure Appl. Algebra 214 (2010), 2017–2026.Google Scholar
Morel, F., The stable ${ \mathbb{A} }^{1} $-connectivity theorems, K-Theory 35 (2005), 1–68.Google Scholar
Morel, F. and Voevodsky, V., ${ \mathbb{A} }^{1} $-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. (2001), 45–143.Google Scholar
Ojanguren, M., The Witt group and the problem of Lüroth. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research]. ETS Editrice, Pisa, 1990. With an introduction by Inta Bertuccioni.Google Scholar
Orlov, D., Vishik, A. and Voevodsky, V., An exact sequence for ${ K}_{\ast }^{M} / 2$ with applications to quadratic forms, Ann. of Math. (2) 165 (2007), 1–13.CrossRefGoogle Scholar
Peyre, E., Unramified cohomology and rationality problems, Math. Ann. 296 (1993), 247–268.CrossRefGoogle Scholar
Rost, M., Norm varieties and algebraic cobordism, in Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002 (Higher Ed. Press, Beijing, 2002), 77–85.Google Scholar
Saltman, D. J., Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), 71–84.Google Scholar
Saltman, D. J., The Brauer group and the center of generic matrices, J. Algebra 97 (1985), 53–67.Google Scholar
Serre, J.-P., Galois cohomology, in Springer Monographs in Mathematics, English edition (Springer, Berlin, 2002), Translated from the French by Patrick Ion and revised by the author.Google Scholar
Suslin, A. and Joukhovitski, S., Norm varieties, J. Pure Appl. Algebra 206 (2006), 245–276.CrossRefGoogle Scholar
Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, in The Arithmetic and Geometry of Algebraic Cycles, Banff, AB, 1998, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 548 (Kluwer Academic Publishers, Dordrecht, 2000), 117–189.Google Scholar
Voevodsky, V., Motivic cohomology with $ \mathbb{Z} / 2$-coefficients, Publ. Math. Inst. Hautes Études Sci. (2003), 59–104.CrossRefGoogle Scholar
Voevodsky, V., On motivic cohomology with $ \mathbb{Z} / l$-coefficients, Ann. of Math. (2) 174 (2011), 401–438.CrossRefGoogle Scholar