Nous établissons la conjecture de Hodge entière réelle pour les 1-cycles pour diverses classes de solides uniréglés (fibrés en coniques, solides de Fano sans points réels, certaines fibrations en del Pezzo) et pour les fibrés en coniques sur des bases de dimension supérieure satisfaisant elles-mêmes la conjecture de Hodge entière réelle pour les 1-cycles. De plus, nous montrons que les solides rationnellement connexes sur les corps réels clos non archimédiens ne vérifient pas en général la conjecture de Hodge entière réelle et que sur de tels corps, le théorème EPT de Bröcker reste vrai pour les surfaces simplement connexes de genre géométrique nul mais tombe en défaut pour certaines surfaces K3.
We establish the real integral Hodge conjecture for -cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for -cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Bröcker’s EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some K3 surfaces.
Accepté le :
Publié le :
DOI : 10.5802/jep.120
Keywords: Real algebraic geometry, integral Hodge conjecture, real closed fields
Mot clés : Géométrie algébrique réelle, conjecture de Hodge entière, corps réels clos
@article{JEP_2020__7__373_0, author = {Benoist, Olivier and Wittenberg, Olivier}, title = {On the integral {Hodge} conjecture for real varieties, {II}}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {373--429}, publisher = {Ecole polytechnique}, volume = {7}, year = {2020}, doi = {10.5802/jep.120}, zbl = {07179024}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.120/} }
TY - JOUR AU - Benoist, Olivier AU - Wittenberg, Olivier TI - On the integral Hodge conjecture for real varieties, II JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 373 EP - 429 VL - 7 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.120/ DO - 10.5802/jep.120 LA - en ID - JEP_2020__7__373_0 ER -
%0 Journal Article %A Benoist, Olivier %A Wittenberg, Olivier %T On the integral Hodge conjecture for real varieties, II %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 373-429 %V 7 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.120/ %R 10.5802/jep.120 %G en %F JEP_2020__7__373_0
Benoist, Olivier; Wittenberg, Olivier. On the integral Hodge conjecture for real varieties, II. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 373-429. doi : 10.5802/jep.120. http://www.numdam.org/articles/10.5802/jep.120/
[AK77] Foundations of the theory of Fano schemes, Compositio Math., Volume 34 (1977) no. 1, pp. 3-47 | Numdam | MR | Zbl
[AK85] A resolution theorem for homology cycles of real algebraic varieties, Invent. math., Volume 79 (1985) no. 3, pp. 589-601 | DOI | MR | Zbl
[AK88] Polynomial equations of immersed surfaces, Pacific J. Math., Volume 131 (1988) no. 2, pp. 209-217 | DOI | MR | Zbl
[Art73] Faisceaux constructibles, cohomologie d’une courbe algébrique, Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4), Tome 3 (Lect. Notes in Math.), Volume 305, Springer-Verlag, Berlin, 1973 (Exp. IX) | Zbl
[Aud91] The topology of torus actions on symplectic manifolds, Progress in Math., 93, Birkhäuser Verlag, Basel, 1991 | MR | Zbl
[BCR98] Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), 36, Springer-Verlag, Berlin, 1998 | MR
[Ben19] The period-index problem for real surfaces, Publ. Math. Inst. Hautes Études Sci., Volume 130 (2019), pp. 63-110 | DOI | MR | Zbl
[BK03] On approximation of smooth submanifolds by nonsingular real algebraic subvarieties, Ann. Sci. École Norm. Sup. (4), Volume 36 (2003) no. 5, pp. 685-690 | DOI | Numdam | MR | Zbl
[BLR90] Néron models, Ergeb. Math. Grenzgeb. (3), 21, Springer-Verlag, Berlin, 1990 | Zbl
[Bru87] Quotient spaces for semialgebraic equivalence relations, Math. Z., Volume 195 (1987) no. 1, pp. 69-78 | DOI | MR | Zbl
[Brö80] Reelle Divisoren, Arch. Math. (Basel), Volume 35 (1980), pp. 140-143 | DOI | MR | Zbl
[BT82] Remarks and counterexamples in the theory of real algebraic vector bundles and cycles, Real algebraic geometry and quadratic forms (Rennes, 1981) (Lect. Notes in Math.), Volume 959, Springer, Berlin, 1982, pp. 198-211 | MR | Zbl
[BW18] On the integral Hodge conjecture for real varieties, I, 2018 (à paraître dans Invent. math.) | arXiv
[Cas12] On the Picard number of divisors in Fano manifolds, Ann. Sci. École Norm. Sup. (4), Volume 45 (2012) no. 3, pp. 363-403 | DOI | Numdam | MR | Zbl
[CK90] Model theory, Studies in Logic and the Foundations of Math., 73, North-Holland Publishing Co., Amsterdam, 1990 | Zbl
[Cle83] Double solids, Adv. in Math., Volume 47 (1983) no. 2, pp. 107-230 | DOI | MR | Zbl
[Com12] Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale, Math. Ann., Volume 73 (1912) no. 1, pp. 1-72 | DOI | MR | Zbl
[Con01] On numbers and games, A K Peters, Ltd., Natick, MA, 2001 | Zbl
[Cor77] Points algébriques sur les surfaces de del Pezzo, C. R. Acad. Sci. Paris Sér. A-B, Volume 284 (1977) no. 24, p. A1531-A1534 | MR | Zbl
[CR82] La topologie du spectre réel, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981) (Contemp. Math.), Volume 8, American Mathematical Society, Providence, R.I., 1982, pp. 27-59 | DOI | Zbl
[CT92] Real rational surfaces without a real point, Arch. Math. (Basel), Volume 58 (1992) no. 4, pp. 392-396 | DOI | MR | Zbl
[CT96] Groupes linéaires sur les corps de fonctions de courbes réelles, J. reine angew. Math., Volume 474 (1996), pp. 139-167 | MR | Zbl
[CTC79] L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques, Compositio Math., Volume 39 (1979) no. 3, pp. 301-332 | Zbl
[CTSD94] Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. reine angew. Math., Volume 453 (1994), pp. 49-112 | MR | Zbl
[CTV12] Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J., Volume 161 (2012) no. 5, pp. 735-801 | DOI | Zbl
[dBJ74] Une propriété de commutation au changement de base des images directes supérieures du faisceau structural, C. R. Acad. Sci. Paris Sér. A-B, Volume 279 (1974), pp. 745-747 | Zbl
[Del71] Théorie de Hodge, II, Publ. Math. Inst. Hautes Études Sci., Volume 40 (1971), pp. 5-57 | DOI | Numdam | Zbl
[Del73] La formule de dualité globale, Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4), Tome 3 (Lect. Notes in Math.), Volume 305, Springer-Verlag, Berlin, 1973 (Exp. XVIII) | Zbl
[Del74] Théorie de Hodge, III, Publ. Math. Inst. Hautes Études Sci., Volume 44 (1974), pp. 5-77 | DOI | Numdam | Zbl
[Del91] Homology of locally semialgebraic spaces, Lect. Notes in Math., 1484, Springer-Verlag, Berlin, 1991 | MR | Zbl
[Dim85] Monodromy and Betti numbers of weighted complete intersections, Topology, Volume 24 (1985) no. 3, pp. 369-374 | DOI | Zbl
[DK81] Semialgebraic topology over a real closed field II: Basic theory of semialgebraic spaces, Math. Z., Volume 178 (1981) no. 2, pp. 175-213 | DOI | MR | Zbl
[DK82] On the homology of algebraic varieties over real closed fields, J. reine angew. Math., Volume 335 (1982), pp. 122-163 | MR | Zbl
[DK84] Separation, retractions and homotopy extension in semialgebraic spaces, Pacific J. Math., Volume 114 (1984) no. 1, pp. 47-71 | DOI | MR | Zbl
[DK85] Locally semialgebraic spaces, Lect. Notes in Math., 1173, Springer-Verlag, Berlin, 1985 | MR | Zbl
[DN14] On the Picard number of singular Fano varieties, Internat. Math. Res. Notices (2014) no. 4, pp. 955-990 | DOI | MR | Zbl
[Duc98] L’obstruction de réciprocité à l’existence de points rationnels pour certaines variétés sur le corps des fonctions d’une courbe réelle, J. reine angew. Math., Volume 504 (1998), pp. 73-114 | DOI | MR | Zbl
[ELP73] On some Hasse principles over formally real fields, Math. Z., Volume 134 (1973), pp. 291-301 | DOI | MR | Zbl
[ELW15] Index of varieties over Henselian fields and Euler characteristic of coherent sheaves, J. Algebraic Geom., Volume 24 (2015) no. 4, pp. 693-718 | DOI | MR | Zbl
[EW16] On the cycle class map for zero-cycles over local fields, Ann. Sci. École Norm. Sup. (4), Volume 49 (2016) no. 2, pp. 483-520 (with an appendix by Spencer Bloch) | DOI | MR | Zbl
[FK10] Topology of real cubic fourfolds, J. Topology, Volume 3 (2010) no. 1, pp. 1-28 | DOI | MR | Zbl
[FK19] First homology of a real cubic is generated by lines, 2019 | arXiv
[Ful98] Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1998 | MR | Zbl
[GH85] On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann., Volume 271 (1985) no. 1, pp. 31-51 | DOI | MR | Zbl
[Gro57] Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2), Volume 9 (1957), pp. 119-221 | Zbl
[Gro61] Techniques de construction et théorèmes d’existence en géométrie algébrique, IV: les schémas de Hilbert, Séminaire Bourbaki, Volume 6, Société Mathématique de France, Paris, 1961, pp. 249-276 Exp. no. 221 (new ed. 1995) | Zbl
[Gro65] Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, II, Publ. Math. Inst. Hautes Études Sci., Volume 24 (1965) | Zbl
[Gro66] Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, III, Publ. Math. Inst. Hautes Études Sci., Volume 28 (1966) | Zbl
[Gro68] Le groupe de Brauer I, II, III, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 46-188 | Zbl
[Hir68] Smoothing of algebraic cycles of small dimensions, Amer. J. Math., Volume 90 (1968), pp. 1-54 | Zbl
[Hir94] Differential topology, Graduate Texts in Math., 33, Springer-Verlag, New York, 1994 | DOI | MR | Zbl
[IP99] Fano varieties, Algebraic geometry, V (Encyclopaedia Math. Sci.), Volume 47, Springer, Berlin, 1999 | MR
[IS88] Rational and homological equivalence for real cycles, Invent. math., Volume 94 (1988) no. 2, pp. 307-316 | MR | Zbl
[Isk79] Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, Volume 12, VINITI, Moscow, 1979, pp. 59-157 | DOI | MR | Zbl
[KK98] Complex algebraic varieties: periods of integrals and Hodge structures, Algebraic geometry, III (Encyclopaedia Math. Sci.), Volume 36, Springer, Berlin, 1998 | Zbl
[KM16] Approximating curves on real rational surfaces, J. Algebraic Geom., Volume 25 (2016) no. 3, pp. 549-570 | DOI | Zbl
[Kne76] On algebraic curves over real closed fields. I, Math. Z., Volume 150 (1976) no. 1, pp. 49-70 | DOI | MR | Zbl
[Kol86] Higher direct images of dualizing sheaves I, Ann. of Math. (2), Volume 123 (1986) no. 1, pp. 11-42 | DOI | Zbl
[Kol93] Shafarevich maps and plurigenera of algebraic varieties, Invent. math., Volume 113 (1993) no. 1, pp. 177-215 | DOI | MR | Zbl
[Kol96] Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996 | DOI | Zbl
[Lam81] The topology of complex projective varieties after S. Lefschetz, Topology, Volume 20 (1981) no. 1, pp. 15-51 | MR
[Lan52] On quasi algebraic closure, Ann. of Math. (2), Volume 55 (1952), pp. 373-390 | DOI | MR | Zbl
[Lan53] The theory of real places, Ann. of Math. (2), Volume 57 (1953) no. 2, pp. 378-391 | DOI | MR | Zbl
[Man66] Rational surfaces over perfect fields, Publ. Math. Inst. Hautes Études Sci., Volume 30 (1966), pp. 55-113 | DOI | MR | Zbl
[Mic08] Topics in differential geometry, Graduate Studies in Math., 93, American Mathematical Society, Providence, RI, 2008, xii+494 pages | Numdam | Zbl
[MM81] Classification of Fano -folds with , manuscripta math., Volume 36 (1981) no. 2, pp. 147-162 | MR | Zbl
[MM83] On Fano -folds with , Algebraic varieties and analytic varieties (Tokyo, 1981) (Adv. Stud. Pure Math.), Volume 1, North-Holland, Amsterdam, 1983, pp. 101-129
[Oes19] Conic bundles and iterated root stacks, European J. Math., Volume 5 (2019) no. 2, pp. 518-527 | DOI
[PS08] Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3), 52, Springer-Verlag, Berlin, 2008 | DOI | MR | Zbl
[PS16] Degree 3 cohomology of function fields of surfaces, Internat. Math. Res. Notices (2016) no. 14, pp. 4341-4374 | MR | Zbl
[Ray70] Spécialisation du foncteur de Picard, Publ. Math. Inst. Hautes Études Sci., Volume 38 (1970), pp. 27-76 | DOI | MR | Zbl
[Sar82] On conic bundle structures, Izv. Akad. Nauk Armjan. SSR Ser. Mat., Volume 46 (1982) no. 2, pp. 371-408 | DOI | Numdam | MR | Zbl
[Sch94] Real and étale cohomology, Lect. Notes in Math., 1588, Springer-Verlag, Berlin, 1994 | MR | Zbl
[Sch95] Purity theorems for real spectra and applications, Real analytic and algebraic geometry (Trento, 1992), de Gruyter, Berlin, 1995, pp. 229-250 | MR | Zbl
[Sch96] Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one, Invent. math., Volume 125 (1996) no. 2, pp. 307-365 | MR | Zbl
[SD72] Rational points on del Pezzo surfaces of degree , Algebraic geometry (Oslo 1970), Wolters-Noordhoff, Groningen, 1972, pp. 287-290 | DOI | MR | Zbl
[SGA1] Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques, 3, Société Mathématique de France, Paris, 2003
[She19] Rationality, universal generation and the integral Hodge conjecture, Geom. Topol., Volume 23 (2019) no. 6, pp. 2861-2898 | DOI | MR | Zbl
[Sil89] Real algebraic surfaces, Lect. Notes in Math., 1392, Springer-Verlag, Berlin, 1989, x+215 pages | MR
[vH00] Algebraic cycles and topology of real algebraic varieties, CWI Tract, 129, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2000 | MR | Zbl
[Voi02] Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, 10, Société Mathématique de France, Paris, 2002 | Zbl
[Voi06] On integral Hodge classes on uniruled or Calabi-Yau threefolds, Moduli spaces and arithmetic geometry (Adv. Stud. Pure Math.), Volume 45, Math. Soc. Japan, Tokyo, 2006, pp. 43-73 | DOI | MR | Zbl
[Wal16] Differential topology, Cambridge Studies in Advanced Math., 156, Cambridge University Press, Cambridge, 2016 | MR | Zbl
[Wei94] An introduction to homological algebra, Cambridge Studies in Advanced Math., 38, Cambridge University Press, Cambridge, 1994 | MR | Zbl
[Wit34] Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper, J. reine angew. Math., Volume 171 (1934), pp. 4-11 | Zbl
[Wit37] Theorie der quadratischen Formen in beliebigen Körpern, J. reine angew. Math., Volume 176 (1937), pp. 31-44 | Zbl
Cité par Sources :