Geometry & Topology

On the Hodge conjecture for $q$–complete manifolds

Franc Forstnerič, Jaka Smrekar, and Alexandre Sukhov

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Abstract

A complex manifold X of dimension n is said to be q–complete for some q {1,,n} if it admits a smooth exhaustion function whose Levi form has at least n q + 1 positive eigenvalues at every point; thus, 1–complete manifolds are Stein manifolds. Such an X is necessarily noncompact and its highest-dimensional a priori nontrivial cohomology group is Hn+q1(X; ). In this paper we show that if q < n, n + q 1 is even, and X has finite topology, then every cohomology class in Hn+q1(X; ) is Poincaré dual to an analytic cycle in X consisting of proper holomorphic images of the ball. This holds in particular for the complement X = n A of any complex projective manifold A defined by q < n independent equations. If X has infinite topology, then the same holds for elements of the group n+q1(X; ) = limjHn+q1(Mj; ), where {Mj}j is an exhaustion of X by compact smoothly bounded domains. Finally, we provide an example of a quasi-projective manifold with a cohomology class which is analytic but not algebraic.

Article information

Source
Geom. Topol., Volume 20, Number 1 (2016), 353-388.

Dates
Received: 9 April 2014
Revised: 6 April 2015
Accepted: 8 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858928

Digital Object Identifier
doi:10.2140/gt.2016.20.353

Mathematical Reviews number (MathSciNet)
MR3470716

Zentralblatt MATH identifier
06553431

Subjects
Primary: 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture 32F10: $q$-convexity, $q$-concavity
Secondary: 32E10: Stein spaces, Stein manifolds 32J25: Transcendental methods of algebraic geometry [See also 14C30]

Keywords
Hodge conjecture complex analytic cycle $q$–complete manifold Stein manifold Poincaré–Lefschetz duality

Citation

Forstnerič, Franc; Smrekar, Jaka; Sukhov, Alexandre. On the Hodge conjecture for $q$–complete manifolds. Geom. Topol. 20 (2016), no. 1, 353--388. doi:10.2140/gt.2016.20.353. https://projecteuclid.org/euclid.gt/1510858928


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References

  • A Andreotti, T Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. 69 (1959) 713–717
  • A Andreotti, R Narasimhan, A topological property of Runge pairs, Ann. of Math. 76 (1962) 499–509
  • A Andreotti, F Norguet, Cycles of algebraic manifolds and $\partial \bar \partial $–cohomology, Ann. Scuola Norm. Sup. Pisa 25 (1971) 59–114
  • M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math. 3, Amer. Math. Soc. (1961) 7–38
  • M F Atiyah, F Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962) 25–45
  • E Ballico, F Catanese, C Ciliberto (editors), Classification of irregular varieties: minimal models and abelian varieties, Lecture Notes in Mathematics 1515, Springer, Berlin (1992)
  • W Barth, Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum, Math. Ann. 187 (1970) 150–162
  • G E Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer, New York (1993)
  • V M Buhštaber, Modules of differentials of the Atiyah–Hirzebruch spectral sequence, II, Mat. Sb. 83 (1970) 61–76 In Russian; translated in Math. USSR-Sb. 12 (1970) 59–75
  • M Colţoiu, $q$–convexity: a survey, from: “Complex analysis and geometry”, (V Ancona, E Ballico, R M Mirò-Roig, A Silva, editors), Pitman Res. Notes Math. Ser. 366, Longman, Harlow (1997) 83–93
  • M Cornalba, P Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975) 1–106
  • P Deligne, The Hodge conjecture, from: “The millennium prize problems”, (J Carlson, A Jaffe, A Wiles, editors), Clay Math. Inst., Cambridge, MA (2006) 45–53
  • J-P Demailly, Cohomology of $q$–convex spaces in top degrees, Math. Z. 204 (1990) 283–295
  • J-P Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics 1, International Press, Somerville, MA (2012)
  • S Donaldson, On the existence of symplectic submanifolds, from: “Contact and symplectic geometry”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 307–310
  • S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666–705
  • A Dor, Immersions and embeddings in domains of holomorphy, Trans. Amer. Math. Soc. 347 (1995) 2813–2849
  • A Douady, Cycles analytiques (d'après M F Atiyah et F Hirzebruch), from: “Séminaire Bourbaki $1961/1962$ (Exposé 223)”, W. A. Benjamin, Amsterdam (1966) Reprinted as pp 5–26 in Séminaire Bourbaki 7, Soc. Math. France, Paris, 1995
  • B Drinovec Drnovšek, F Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007) 203–253
  • B Drinovec-Drnovšek, F Forstnerič, Approximation of holomorphic mappings on strongly pseudoconvex domains, Forum Math. 20 (2008) 817–840
  • B Drinovec Drnovšek, F Forstnerič, Strongly pseudoconvex domains as subvarieties of complex manifolds, Amer. J. Math. 132 (2010) 331–360
  • O Forster, Plongements des variétés de Stein, Comment. Math. Helv. 45 (1970) 170–184
  • F Forstnerič, Noncritical holomorphic functions on Stein manifolds, Acta Math. 191 (2003) 143–189
  • F Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006) 239–270
  • F Forstnerič, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. 56, Springer, Heidelberg (2011)
  • F Forstnerič, J Globevnik, Discs in pseudoconvex domains, Comment. Math. Helv. 67 (1992) 129–145
  • H Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958) 263–273
  • H Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958) 460–472
  • H Grauert, Une notion de dimension cohomologique dans la théorie des espaces complexes, Bull. Soc. Math. France 87 (1959) 341–350
  • H Grauert, Theory of $q$–convexity and $q$–concavity, from: “Several complex variables, VII”, (H Grauert, T Peternell, R Remmert, editors), Encyclopaedia Math. Sci. 74, Springer, Berlin (1994) 259–284
  • M L Green, Infinitesimal methods in Hodge theory, from: “Algebraic cycles and Hodge theory”, (A Albano, F Bardelli, editors), Lecture Notes in Math. 1594, Springer, Berlin (1994) 1–92
  • P Griffiths, J Harris, On the Noether–Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985) 31–51
  • A Grothendieck, Hodge's general conjecture is false for trivial reasons, Topology 8 (1969) 299–303
  • H A Hamm, Zur Homotopietyp Steinscher Räume, J. Reine Angew. Math. 338 (1983) 121–135
  • H A Hamm, Zum Homotopietyp $q$–vollständiger Räume, J. Reine Angew. Math. 364 (1986) 1–9
  • G M Henkin, J Leiterer, Andreotti–Grauert theory by integral formulas, Progress in Mathematics 74, Birkhäuser, Boston (1988)
  • W V D Hodge, The topological invariants of algebraic varieties, from: “Proceedings of the International Congress of Mathematicians, I”, Amer. Math. Soc. (1952) 182–192
  • C U Jensen, Les foncteurs dérivés de $\varprojlim$ et leurs applications en théorie des modules, Lecture Notes in Mathematics 254, Springer, Berlin (1972)
  • B J öricke, Envelopes of holomorphy and holomorphic discs, Invent. Math. 178 (2009) 73–118
  • S Kaliman, M Zaĭdenberg, A tranversality theorem for holomorphic mappings and stability of Eisenman–Kobayashi measures, Trans. Amer. Math. Soc. 348 (1996) 661–672
  • K Kodaira, D C Spencer, Divisor class groups on algebraic varieties, Proc. Nat. Acad. Sci. USA 39 (1953) 872–877
  • W S Massey, Singular homology theory, Graduate Texts in Mathematics 70, Springer, New York (1980)
  • J Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962) 337–341
  • T Ohsawa, Completeness of noncompact analytic spaces, Publ. Res. Inst. Math. Sci. 20 (1984) 683–692
  • M Peternell, $q$–completeness of subsets in complex projective space, Math. Z. 195 (1987) 443–450
  • M Schneider, Über eine Vermutung von Hartshorne, Math. Ann. 201 (1973) 221–229
  • G Sorani, Omologia degli spazi $q$–pseudoconvessi, Ann. Scuola Norm. Sup. Pisa 16 (1962) 299–304
  • C Soulé, C Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198 (2005) 107–127
  • E H Spanier, Algebraic topology, Springer, New York (1981) Corrected reprint
  • B Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997) 467–493
  • S Trivedi, Stratified transversality of holomorphic maps, Internat. J. Math. 24 (2013) 1350106, 12
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)