Soit $R$ le corps des séries de Puiseux réelles. C’est un corps réel clos. On construit les premiers exemples d’intersections lisses de deux quadriques dans $\mathbb{P}_R^5$ et d’hypersurfaces cubiques lisses dans $\mathbb{P}_R^4$ qui ne sont pas stablement rationnelles mais pour lesquelles l’espace $X(R)$ des $R$-points est semi-algébriquement connexe. La question de construire de tels exemples sur le corps des réels $\mathbb{R}$ reste ouverte.
Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably rational but for which the space $X(R)$ of $R$-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers $\mathbb{R}$ remains open.
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Mots-clés : Rationalité, connexité réelle, géométrie semi-algébrique, spécialisation, cohomologie non ramifiée, formes quadratiques, groupes de Chow
Keywords: Rationality, real connectedness, semi-algebraic geometry, specialization, unramified cohomology, quadratic forms, Chow groups
Jean-Louis Colliot-Thélène 1 ; Alena Pirutka 2 ; Federico Scavia 3
CC-BY 4.0
Jean-Louis Colliot-Thélène; Alena Pirutka; Federico Scavia. Variétés réelles semi-algébriquement connexes non stablement rationnelles. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1061-1098. doi: 10.5802/jep.343
@article{JEP_2026__13__1061_0,
author = {Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka and Federico Scavia},
title = {Vari\'et\'es r\'eelles semi-alg\'ebriquement connexes non stablement rationnelles},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1061--1098},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.343},
language = {fr},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.343/}
}
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[ACTP17] - “Universal unramified cohomology of cubic fourfolds containing a plane”, in Brauer groups and obstruction problems, Progress in Math., vol. 320, Birkhäuser/Springer, Cham, 2017, p. 29-55 | DOI | Zbl
[Ara75] - “Cohomologische Invarianten quadratischer Formen”, J. Algebra 36 (1975) no. 3, p. 448-491 | DOI | Zbl | MR
[BCR87] - Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. (3), vol. 12, Springer-Verlag, Berlin, 1987 | Zbl | MR
[BCR98] - Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), vol. 36, Springer-Verlag, Berlin, 1998, Translated from the 1987 French original, Revised by the authors | DOI | Zbl
[Blo86] - “Algebraic cycles and higher -theory”, Adv. Math. 61 (1986) no. 3, p. 267-304 | DOI | Zbl
[BP26] - “On the rationality of some real threefolds”, J. Inst. Math. Jussieu 25 (2026) no. 3, p. 1555-1577 | DOI | Zbl | MR
[BW20a] - “The Clemens-Griffiths method over non-closed fields”, Algebraic Geom. 7 (2020) no. 6, p. 696-721 | DOI | Zbl | MR
[BW20b] - “On the integral Hodge conjecture for real varieties, I”, Invent. math. 222 (2020) no. 1, p. 1-77 | DOI | Zbl | MR
[BW20c] - “On the integral Hodge conjecture for real varieties, II”, J. Éc. polytech. Math. 7 (2020), p. 373-429 | DOI | Numdam | Zbl | MR
[BW23] - “Intermediate Jacobians and rationality over arbitrary fields”, Ann. Sci. École Norm. Sup. (4) 56 (2023) no. 4, p. 1029-1084, Erratum : Ibid., 58 (2025), no. 3, p. 829–830 | DOI | Zbl | MR
[Com12] - “Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale.”, Math. Ann. 73 (1912), p. 1-72 | DOI | Zbl | MR
[CS92] - “Nash triviality in families of Nash manifolds”, Invent. math. 108 (1992) no. 2, p. 349-368 | DOI | Zbl | MR
[CT95] - “Birational invariants, purity and the Gersten conjecture”, in -theory and algebraic geometry : connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, Part 1, American Mathematical Society, Providence, RI, 1995, p. 1-64 | DOI | Zbl
[CTP90] - “Real components of algebraic varieties and étale cohomology”, Invent. math. 101 (1990) no. 1, p. 81-99 | DOI | Zbl | MR
[CTP16] - “Hypersurfaces quartiques de dimension 3 : non-rationalité stable”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 2, p. 371-397 | DOI | Numdam | Zbl | MR
[CTP24] - “Certaines fibrations en surfaces quadriques réelles”, 2024, À paraître dans Comment. Math. Helv. | arXiv | Zbl
[CTS93] - “Groupe de Chow des zéro-cycles sur les fibrés en quadriques”, -Theory 7 (1993) no. 5, p. 477-500 | DOI | Zbl | MR
[CTS21] - The Brauer-Grothendieck group, Ergeb. Math. Grenzgeb. (3), vol. 71, Springer, Cham, 2021 | DOI | Zbl | MR
[CTSSD87] - “Intersections of two quadrics and Châtelet surfaces. I”, J. reine angew. Math. 373 (1987), p. 37-107 | DOI | Zbl
[CTZ26] - “Rationality of singular cubic threefolds over ”, Adv. Math. 487 (2026), article ID 110756, 27 pages | DOI | MR | Zbl
[DK81a] - “Semialgebraic topology over a real closed field. I. Paths and components in the set of rational points of an algebraic variety”, Math. Z. 177 (1981) no. 1, p. 107-129 | DOI | MR | Zbl
[DK81b] - “Semialgebraic topology over a real closed field. II. Basic theory of semialgebraic spaces”, Math. Z. 178 (1981) no. 2, p. 175-213 | DOI | MR | Zbl
[Ful75] - “Rational equivalence on singular varieties”, Publ. Math. Inst. Hautes Études Sci. (1975) no. 45, p. 147-167 | DOI | Numdam | MR | Zbl
[Ful98] - Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1998 | DOI | Zbl
[Har77] - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977 | DOI | MR | Zbl
[HKT22] - “Rationality of even-dimensional intersections of two real quadrics”, Comment. Math. Helv. 97 (2022) no. 1, p. 183-207 | DOI | MR | Zbl
[HT21] - “Rationality of complete intersections of two quadrics over nonclosed fields”, Enseign. Math. (2) 67 (2021) no. 1-2, p. 1-44, with an appendix by Jean-Louis Colliot-Thélène | DOI | MR | Zbl
[Kah08] - Formes quadratiques sur un corps, Cours Spécialisés, vol. 15, Société Mathématique de France, Paris, 2008 | MR | Zbl
[KRS98] - “Unramified cohomology of quadrics. I”, Amer. J. Math. 120 (1998) no. 4, p. 841-891 | DOI | Zbl | MR
[KS01] - “Unramified cohomology of quadrics. II”, Duke Math. J. 106 (2001) no. 3, p. 449-484 | DOI | MR | Zbl
[Lam05] - Introduction to quadratic forms over fields, Graduate Studies in Math., vol. 67, American Mathematical Society, Providence, RI, 2005 | DOI | MR | Zbl
[Mer81] - “On the norm residue symbol of degree ”, Dokl. Akad. Nauk SSSR 261 (1981) no. 3, p. 542-547 | MR
[Mer08] - “Unramified elements in cycle modules”, J. London Math. Soc. (2) 78 (2008) no. 1, p. 51-64 | DOI | MR | Zbl
[Ros96] - “Chow groups with coefficients”, Doc. Math. 1 (1996), p. 319-393 | DOI | MR | Zbl
[Sch94] - Real and étale cohomology, Lect. Notes in Math., vol. 1588, Springer-Verlag, Berlin, 1994 | DOI | MR | Zbl
[Sch96] - “Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one”, Invent. math. 125 (1996) no. 2, p. 307-365 | DOI | MR | Zbl
[Sch21] - “Unramified cohomology, algebraic cycles and rationality”, in Rationality of varieties, Progress in Math., vol. 342, Birkhäuser/Springer, Cham, 2021, p. 345-388 | DOI | Zbl
[Sch24] - A course in real algebraic geometry—positivity and sums of squares, Graduate Texts in Math., vol. 303, Springer, Cham, 2024 | DOI | MR | Zbl
[SGA4II] - Théorie des topos et cohomologie étale des schémas. Tome 2 (M. Artin, A. Grothendieck & J. L. Verdier, eds.), Lect. Notes in Math., vol. 270, Springer-Verlag, Berlin–New York, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), avec des contributions de N. Bourbaki, P. Deligne et B. Saint-Donat | DOI
[Wit37] - “Theorie der quadratischen Formen in beliebigen Körpern”, J. reine angew. Math. 176 (1937), p. 31-44 | DOI | MR | Zbl
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