Abstract
We formulate the “real integral Hodge conjecture”, a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi–Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel–Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these problems are obtained along the way.
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Notes
We mean Calabi–Yau in the sense that \(K_X\simeq {{\mathscr {O}}}_X\) and \(H^1(X,{{\mathscr {O}}}_X)=H^2(X,{{\mathscr {O}}}_X)=0\). Totaro has announced a proof of the integral Hodge conjecture for complex projective threefolds X such that \(K_X \simeq {{\mathscr {O}}}_X\). Abelian threefolds are dealt with in [58, Corollary 3.1.9]. The hypothesis that \(K_X \simeq {{\mathscr {O}}}_X\) cannot be weakened to \(K_X\) being torsion in view of [21, Theorem 0.1].
References
Atiyah, M.F., Hirzebruch, F.: Cohomologie-Operationen und charakteristische Klassen. Math. Z. 77, 149–187 (1961)
Atiyah, M.F., Hirzebruch, F.: Analytic cycles on complex manifolds. Topology 1, 25–45 (1962)
Akbulut, S., King, H.: Submanifolds and homology of nonsingular real algebraic varieties. Am. J. Math. 107(1), 45–83 (1985)
Akbulut, S., King, H.: Polynomial equations of immersed surfaces. Pac. J. Math. 131(2), 209–217 (1988)
Akbulut, S., King, H.: Transcendental submanifolds of \(\mathbf{R}^n\). Comment. Math. Helv. 68(2), 308–318 (1993)
Araujo, C., Kollár, J.: Rational curves on varieties. In: Higher Dimensional Varieties and Rational Points (Budapest, 2001), Bolyai Soc. Math. Stud., vol. 12, pp. 13–68. Springer, Berlin (2003)
Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Cambridge (1993)
Ballico, E., Catanese, F., Ciliberto, C.: Trento examples. In: Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Math., vol. 1515, pp. 134–139. Springer, Berlin (1992)
Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Ergeb. Math. Grenzgeb. (3), vol. 36, Springer, Berlin (1998). Translated from the 1987 French original, revised by the authors
Benedetti, R., Dedò, M.: Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism. Compos. Math. 53(2), 143–151 (1984). erratum in 55, no. 3 (1985), p. 400
Benoist, O.: On Hilbert’s 17th problem in low degree. Algebra Number Theory 11(4), 929–959 (2017)
Berthelot, P.: Quelques calculs de groupes \(K\), Exp. IX, Théorie des intersections et théorème de Riemann–Roch, Séminaire de géométrie algébrique du Bois-Marie 1966–1967 (SGA 6). In: Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)
Borel, A., Haefliger, A.: La classe d’homologie fondamentale d’un espace analytique. Bull. Soc. Math. France 89, 461–513 (1961)
Bochnak, J., Kucharz, W.: Algebraic models of smooth manifolds. Invent. math. 97(3), 585–611 (1989)
Bochnak, J., Kucharz, W.: On homology classes represented by real algebraic varieties. In: Singularities Symposium—Łojasiewicz 70 (Kraków, 1996; Warsaw, 1996), Banach Center Publ., vol. 44, pp. 21–35. Polish Acad. Sci. Inst. Math., Warsaw (1998)
Bochnak, J., Kucharz, W.: On approximation of smooth submanifolds by nonsingular real algebraic subvarieties. Ann. Sci. École Norm. Sup. (4) 36(5), 685–690 (2003)
Bloch, S.: Torsion algebraic cycles and a theorem of Roitman. Compos. Math. 39(1), 107–127 (1979)
Bloch, S.: Lectures on Algebraic Cycles. Duke University Mathematics Series, IV, Mathematics Department, Duke University, Durham (1980)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models, Ergeb. Math. Grenzgeb. (3), vol. 21. Springer-Verlag, Berlin (1990)
Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. École Norm. Sup. 4(7), 181–201 (1974)
Benoist, O., Ottem, J.C.: Failure of the integral Hodge conjecture for threefolds of Kodaira dimension 0, arXiv:1802.01845, to appear in Comment. Math. Helv
Bredon, G.E.: Sheaf Theory. Graduate Texts in Mathematics, vol. 170, 2nd edn. Springer, New York (1997)
Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1994)
Brumfiel, G.W.: Quotient spaces for semialgebraic equivalence relations. Math. Z. 195(1), 69–78 (1987)
Brumfiel, G.W.: A Hopf fixed point theorem for semi-algebraic maps. In: Real Algebraic Geometry (Rennes, 1991), Lecture Notes in Math., vol. 1524, pp. 163–169. Springer, Berlin (1992)
Borel, A., Serre, J.-P.: Le théorème de Riemann–Roch. Bull. Soc. Math. France 86, 97–136 (1958)
Bloch, S., Srinivas, V.: Remarks on correspondences and algebraic cycles. Am. J. Math. 105(5), 1235–1253 (1983)
Benedetti, R., Tognoli, A.: Remarks and counterexamples in the theory of real algebraic vector bundles and cycles. In: Real Algebraic Geometry and Quadratic Forms (Rennes, 1981), Lecture Notes in Math., vol. 959, pp. 198–211. Springer, Berlin (1982)
Benoist, O., Wittenberg, O.: On the integral Hodge conjecture for real varieties, II. J. de l’École polytechnique — Mathématiques 7, 373–429 (2020)
Colliot-Thélène, J.-L.: Cycles algébriques de torsion et \(K\)-théorie algébrique. In: Arithmetic Algebraic Geometry (Trento, 1991), Lecture Notes in Math., vol. 1553, pp. 1–49. Springer, Berlin (1993)
Colliot-Thélène, J.-L., Hoobler, R.T., Kahn, B.: The Bloch–Ogus–Gabber Theorem, Algebraic \(K\)-theory (Toronto, ON, 1996), Fields Inst. Commun., vol. 16, pp. 31–94. Amer. Math. Soc., Providence, RI (1997)
Colliot-Thélène, J.-L., Madore, D.A.: Surfaces de del Pezzo sans point rationnel sur un corps de dimension cohomologique un. J. Inst. Math. Jussieu 3(1), 1–16 (2004)
Colliot-Thélène, J.-L., Parimala, R.: Real components of algebraic varieties and étale cohomology. Invent. math. 101(1), 81–99 (1990)
Colliot-Thélène, J.-L., Scheiderer, C.: Zero-cycles and cohomology on real algebraic varieties. Topology 35(2), 533–559 (1996)
Colliot-Thélène, J.-L., Sansuc, J.-J., Soulé, C.: Torsion dans le groupe de Chow de codimension deux. Duke Math. J. 50(3), 763–801 (1983)
Colliot-Thélène, J.-L., Voisin, C.: Cohomologie non ramifiée et conjecture de Hodge entière. Duke Math. J. 161(5), 735–801 (2012)
Deligne, P.: La formule de dualité globale, Exp. XVIII. In: Théorie des Topos et Cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4), Tome 3, Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973)
Delfs, H.: The homotopy axiom in semialgebraic cohomology. J. reine angew. Math. 355, 108–128 (1985)
Delfs, H.: Homology of Locally Semialgebraic Spaces. Lecture Notes in Mathematics, vol. 1484. Springer, Berlin (1991)
Delfs, H., Knebusch, M.: Semialgebraic topology over a real closed field I: paths and components in the set of rational points of an algebraic variety. Math. Z. 177(1), 107–129 (1981)
Delfs, H.: Semialgebraic topology over a real closed field II: basic theory of semialgebraic spaces. Math. Z. 178(2), 175–213 (1981)
Delfs, H.: On the homology of algebraic varieties over real closed fields. J. reine angew. Math. 335, 122–163 (1982)
Delfs, H.: Semialgebraic topology over a real closed field. In: Ordered Fields and Real Algebraic Geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, pp. 61–78 Amer. Math. Soc., Providence, R.I. (1982)
Delfs, H.: An introduction to locally semialgebraic spaces. Rocky Mountain J. Math. 14(4), 945–963 (1984). Ordered fields and real algebraic geometry (Boulder, Colo., 1983)
Delfs, H.: Locally Semialgebraic Spaces. Lecture Notes in Mathematics, vol. 1173. Springer, Berlin (1985)
Degtyarev, A., Kharlamov, V.: Topological classification of real Enriques surfaces. Topology 35(3), 711–729 (1996)
dos Santos, P.F., Lima-Filho, P.: Bigraded equivariant cohomology of real quadrics. Adv. Math. 221(4), 1247–1280 (2009)
Eisenbud, D., Harris, J.: 3264 and All That, A Second Course in Algebraic Geometry. Cambridge University Press, Cambridge (2016)
Esnault, H., Levine, M., Wittenberg, O.: Index of varieties over Henselian fields and Euler characteristic of coherent sheaves. J. Algebraic Geom. 24(4), 693–718 (2015)
Edmundo, M.J., Prelli, L.: The six Grothendieck operations on o-minimal sheaves. Math. Z. 294(1–2), 109–160 (2020)
Epstein, D.B.A.: Steenrod operations in homological algebra. Invent. math. 1, 152–208 (1966)
Floris, E.: Fundamental divisors on Fano varieties of index \(n-3\). Geom. Dedicata 162, 1–7 (2013)
Fulton, W.: Intersection Theory. Ergeb. Math. Grenzgeb. (3), vol. 2, 2nd edn. Springer, Berlin (1998)
Geyer, W.-D.: Ein algebraischer Beweis des Satzes von Weichold über reele algebraische Funktionenkörper, Algebraische Zahlentheorie (Ber. Tagung Math. Forschungsinst. Oberwolfach, 1964), pp. 83–98 . Bibliographisches Institut, Mannheim (1967)
Gross, B.H., Harris, J.: Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14(2), 157–182 (1981)
Griffiths, P., Harris, J.: On the Noether–Lefschetz theorem and some remarks on codimension-two cycles. Math. Ann. 271(1), 31–51 (1985)
Godeaux, L.: Sur la construction de surfaces non rationnelles de genres zéro. Acad. R. Belg. Bull. Cl. Sci. (5) 35, 688–693 (1949)
Grabowski, C.: On the integral Hodge conjecture for 3-folds. Ph. D. thesis, Duke University (2004)
Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. (2) 9, 119–221 (1957)
Grothendieck, A.: Le Groupe de Brauer I, II, III, Dix exposés sur la cohomologie des schémas, pp. 46–188. North-Holland, Amsterdam (1968)
Haution, O.: Degree formula for the Euler characteristic. Proc. Am. Math. Soc. 141(6), 1863–1869 (2013)
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I. Ann. Math. (2) 79, 109–203 (1964)
Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, New York (1994). corrected reprint of the 1976 original
Höring, A., Voisin, C.: Anticanonical divisors and curve classes on Fano manifolds. Pure Appl. Math. Q. 7(4), 1371–1393 (2011)
Heller, J., Voineagu, M.: Equivariant semi-topological invariants, Atiyah’s \(KR\)-theory, and real algebraic cycles. Trans. Am. Math. Soc. 364(12), 6565–6603 (2012)
Jannsen, U.: Motivic sheaves and filtrations on Chow groups. In: Motives (Seattle, WA, 1991), Proceedings of the Symposium Pure Math., vol. 55, pp. 245–302. Amer. Math. Soc., Providence, RI (1994)
Kleiman, S., Altman, A.: Bertini theorems for hypersurface sections containing a subscheme. Commun. Algebra 7(8), 775–790 (1979)
Kahn, B.: Construction de classes de Chern équivariantes pour un fibré vectoriel réel. Commun. Algebra 15(4), 695–711 (1987)
Kahn, B.: Deux théorèmes de comparaison en cohomologie étale: applications. Duke Math. J. 69(1), 137–165 (1993)
Karpenko, N.A.: Algebro-geometric invariants of quadratic forms. Algebra i Anal. 2(1), 141–162 (1990)
Katz, N.: Etude cohomologique des pinceaux de Lefschetz, Exp. XVIII. In: Groupes de monodromie en géométrie algébrique, II, Séminaire de géométrie algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, vol. 340. Springer, Berlin-New York (1973)
Knebusch, M.: On algebraic curves over real closed fields. I. Math. Z. 150(1), 49–70 (1976)
Knebusch, M.: On algebraic curves over real closed fields. II. Math. Z. 151(2), 189–205 (1976)
Kollár, J.: Esnault–Levine–Wittenberg indices, preprint (2013). arXiv:1312.3923
Krasnov, V.A.: Characteristic classes of vector bundles on a real algebraic variety. Izv. Akad. Nauk SSSR Ser. Mat. 55(4), 716–746 (1991)
Krasnov, V.A.: On the equivariant Grothendieck cohomology of a real algebraic variety and its application. Izv. Ross. Akad. Nauk Ser. Mat. 58(3), 36–52 (1994)
Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der mathematischen Wissenschaften, vol. 292. Springer, Berlin (1994)
Kucharz, W.: On homology of real algebraic sets. Invent. math. 82(1), 19–25 (1985)
Kucharz, W.: Algebraic equivalence and homology classes of real algebraic cycles. Math. Nachr. 180, 135–140 (1996)
Kucharz, W.: Algebraic equivalence of real divisors. Math. Z. 238(4), 817–827 (2001)
Lewis, J.D.: A Survey of the Hodge Conjecture. CRM Monograph Series, vol. 10, 2nd edn. American Mathematical Society, Providence (1999)
Lichtenbaum, S.: Duality theorems for curves over \(p\)-adic fields. Invent. math. 7, 120–136 (1969)
Mangolte, F.: Cycles algébriques sur les surfaces \(K3\) réelles. Math. Z. 225(4), 559–576 (1997)
Mangolte, F.: Variétés algébriques réelles, Cours spécialisés, vol. 24. Société Mathématique de France, Paris (2017)
Milne, J.S.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)
Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, Annals of Mathematics Studies, No. 76 (1974)
Mangolte, F., van Hamel, J.: Algebraic cycles and topology of real Enriques surfaces. Compos. Math. 110(2), 215–237 (1998)
Panin, I.: Riemann-Roch theorems for oriented cohomology. In: Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, pp. 261–333 Kluwer Acad. Publ., Dordrecht (2004)
Pfister, A.: Zur Darstellung von \(-1\) als Summe von Quadraten in einem Körper. J. Lond. Math. Soc. 40, 159–165 (1965)
Raynaud, M.M.: Modules projectifs universels. Invent. math. 6, 1–26 (1968)
Reid, M.: Surfaces with \(p_{g}=0\), \(K^{2}=2\), preprint (1979). available from the author’s webpage, http://homepages.warwick.ac.uk/~masda/surf/K2=2.pdf
Risler, J.-J.: Sur l’homologie des surfaces algébriques réelles. In: Real Algebraic Geometry and Quadratic Forms (Rennes, 1981), Lecture Notes in Math., vol. 959, pp. 381–385. Springer, Berlin-New York (1982)
Robson, R.: Embedding semi-algebraic spaces. Math. Z. 183(3), 365–370 (1983)
Roĭtman, A.A.: Rational equivalence of 0-cycles. Mat. Sb. (N.S.) 89, 569–585 (1972)
Saito, S.: A global duality theorem for varieties over global fields. In: Algebraic \(K\)-Theory: Connections with Geometry and Topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 279, pp. 425–444. Kluwer Acad. Publ., Dordrecht (1989)
Scheiderer, C.: Real and étale Cohomology. Lecture Notes in Mathematics, vol. 1588. Springer, Berlin (1994)
Scheiderer, C.: Purity theorems for real spectra and applications. In: Real Analytic and Algebraic Geometry (Trento, 1992), pp. 229–250. de Gruyter, Berlin (1995)
Schoen, C.: An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite fields. Math. Ann. 311(3), 493–500 (1998)
Schreieder, S.: Stably irrational hypersurfaces of small slopes. J. Amer. Math. Soc. 32(4), 1171–1199 (2019)
Silhol, R.: A bound on the order of \(H^{(a)}_{n-1}(X,\,{\bf Z}/2)\) on a real algebraic variety. In: Real Algebraic Geometry and Quadratic Forms (Rennes, 1981), Lecture Notes in Math., vol. 959, pp. 443–450 Springer, Berlin-New York (1982)
Silhol, R.: Real Algebraic Surfaces. Lecture Notes in Mathematics, vol. 1392. Springer, Berlin (1989)
Soulé, C., Voisin, C.: Torsion cohomology classes and algebraic cycles on complex projective manifolds. Adv. Math. 198(1), 107–127 (2005)
Teichner, P.: 6-dimensional manifolds without totally algebraic homology. Proc. Am. Math. Soc. 123(9), 2909–2914 (1995)
Totaro, B.: On the integral Hodge and Tate conjectures over a number field. Forum Math. Sigma 1, e4 (2013)
van Hamel, J.: Algebraic Cycles and Topology of Real Algebraic Varieties. CWI Tract, vol. 129, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, Dissertation, Vrije Universiteit Amsterdam, Amsterdam (2000)
van Hamel, J.: Torsion zero-cycles and the Abel–Jacobi map over the real numbers. In: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), CRM Proceedings of the Lecture Notes, vol. 24, pp. 329–359. Amer. Math. Soc., Providence, RI (2000)
Voisin, C.: Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, vol. 10. Société Mathématique de France, Paris (2002)
Voisin, C.: On integral Hodge classes on uniruled or Calabi–Yau threefolds. In: Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math., vol. 45, pp. 43–73. Math. Soc. Japan, Tokyo (2006)
Voisin, C.: Some aspects of the Hodge conjecture. Jpn. J. Math. 2(2), 261–296 (2007)
Voisin, C.: Remarks on curve classes on rationally connected varieties. In: A Celebration of Algebraic Geometry, Clay Math. Proc., vol. 18, pp. 591–599. Amer. Math. Soc., Providence, RI (2013)
Voisin, C.: Chow Rings, Decomposition of the Diagonal, and the Topology of Families. Annals of Mathematics Studies, vol. 187. Princeton University Press, Princeton (2014)
Witt, E.: Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper. J. reine angew. Math. 171, 4–11 (1934)
Acknowledgements
Krasnov and van Hamel were the first to approach algebraic cycles on real varieties in a systematic way through the study of the cycle class map into the equivariant integral singular cohomology of the complex locus. We would like to emphasise the importance of their work to the development of the subject considered in this article. In addition, we thank the referee for their careful work and for many suggestions which helped improve the exposition.
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Benoist, O., Wittenberg, O. On the integral Hodge conjecture for real varieties, I. Invent. math. 222, 1–77 (2020). https://doi.org/10.1007/s00222-020-00965-8
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DOI: https://doi.org/10.1007/s00222-020-00965-8