On the Hodge conjecture for q–complete manifolds

Forstnerič, Franc¹ ; Smrekar, Jaka¹ ; Sukhov, Alexandre²

¹ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia, Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
² Laboratoire Paul Painleve, UFR de Mathematiques, Universite Lille-1, F-59655 Villeneuve d’Ascq Cedex, France

Geometry & topology, Tome 20 (2016) no. 1, pp. 353-388.

Résumé

A complex manifold $X$ of dimension $n$ is said to be $q$–complete for some $q\in \left\{1,\dots ,n\right\}$ 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 $H^{n+q-1}\left(X;\mathbb{Z}\right)$. In this paper we show that if $q<n$, $n+q-1$ is even, and $X$ has finite topology, then every cohomology class in $H^{n+q-1}\left(X;\mathbb{Z}\right)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X=ℂ{\mathbb{P}}^n\setminus A$ 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 ${\mathcal{\mathscr{H}}}^{n+q-1}\left(X;\mathbb{Z}\right)=\underset{j}{lim}{H}^{n+q-1}\left(M_j;\mathbb{Z}\right)$, where ${\left\{M_j\right\}}_{j\in \mathbb{N}}$ 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.

Détail

DOI : 10.2140/gt.2016.20.353

Classification : 14C30, 32F10, 32E10, 32J25
Keywords: Hodge conjecture, complex analytic cycle, $q$–complete manifold, Stein manifold, Poincaré–Lefschetz duality

Bibliographie

[1] A Andreotti, T Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. 69 (1959) 713

[2] A Andreotti, R Narasimhan, A topological property of Runge pairs, Ann. of Math. 76 (1962) 499

[3] A Andreotti, F Norguet, Cycles of algebraic manifolds and ∂∂–cohomology, Ann. Scuola Norm. Sup. Pisa 25 (1971) 59

[4] M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math. 3, Amer. Math. Soc. (1961) 7

[5] M F Atiyah, F Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962) 25

[6] E Ballico, F Catanese, C Ciliberto, editors, Classification of irregular varieties: minimal models and abelian varieties, 1515, Springer (1992)

[7] W Barth, Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum, Math. Ann. 187 (1970) 150

[8] G E Bredon, Topology and geometry, 139, Springer (1993)

[9] V M Buhštaber, Modules of differentials of the Atiyah–Hirzebruch spectral sequence, II, Mat. Sb. 83 (1970) 61

[10] M Colţoiu, q–convexity : a survey, from: "Complex analysis and geometry" (editors V Ancona, E Ballico, R M Mirò-Roig, A Silva), Pitman Res. Notes Math. Ser. 366, Longman, Harlow (1997) 83

[11] M Cornalba, P Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties, Invent. Math. 28 (1975) 1

[12] P Deligne, The Hodge conjecture, from: "The millennium prize problems" (editors J Carlson, A Jaffe, A Wiles), Clay Math. Inst. (2006) 45

[13] J P Demailly, Cohomology of q–convex spaces in top degrees, Math. Z. 204 (1990) 283

[14] J P Demailly, Analytic methods in algebraic geometry, 1, International Press (2012)

[15] S Donaldson, On the existence of symplectic submanifolds, from: "Contact and symplectic geometry" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 307

[16] S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666

[17] A Dor, Immersions and embeddings in domains of holomorphy, Trans. Amer. Math. Soc. 347 (1995) 2813

[18] A Douady, Cycles analytiques (d’après M F Atiyah et F Hirzebruch), from: "Séminaire Bourbaki 1961∕1962 (Exposé 223)", W. A. Benjamin (1966)

[19] B Drinovec Drnovšek, F Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007) 203

[20] B Drinovec-Drnovšek, F Forstnerič, Approximation of holomorphic mappings on strongly pseudoconvex domains, Forum Math. 20 (2008) 817

[21] B Drinovec Drnovšek, F Forstnerič, Strongly pseudoconvex domains as subvarieties of complex manifolds, Amer. J. Math. 132 (2010) 331

[22] O Forster, Plongements des variétés de Stein, Comment. Math. Helv. 45 (1970) 170

[23] F Forstnerič, Noncritical holomorphic functions on Stein manifolds, Acta Math. 191 (2003) 143

[24] F Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006) 239

[25] F Forstnerič, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, 56, Springer (2011)

[26] F Forstnerič, J Globevnik, Discs in pseudoconvex domains, Comment. Math. Helv. 67 (1992) 129

[27] H Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958) 263

[28] H Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958) 460

[29] H Grauert, Une notion de dimension cohomologique dans la théorie des espaces complexes, Bull. Soc. Math. France 87 (1959) 341

[30] H Grauert, Theory of q–convexity and q–concavity, from: "Several complex variables, VII" (editors H Grauert, T Peternell, R Remmert), Encyclopaedia Math. Sci. 74, Springer (1994) 259

[31] M L Green, Infinitesimal methods in Hodge theory, from: "Algebraic cycles and Hodge theory" (editors A Albano, F Bardelli), Lecture Notes in Math. 1594, Springer, Berlin (1994) 1

[32] P Griffiths, J Harris, On the Noether–Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985) 31

[33] A Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969) 299

[34] H A Hamm, Zur Homotopietyp Steinscher Räume, J. Reine Angew. Math. 338 (1983) 121

[35] H A Hamm, Zum Homotopietyp q–vollständiger Räume, J. Reine Angew. Math. 364 (1986) 1

[36] G M Henkin, J Leiterer, Andreotti–Grauert theory by integral formulas, 74, Birkhäuser (1988) 270

[37] W V D Hodge, The topological invariants of algebraic varieties, from: "Proceedings of the International Congress of Mathematicians, I", Amer. Math. Soc. (1952) 182

[38] C U Jensen, Les foncteurs dérivés de et leurs applications en théorie des modules, 254, Springer (1972)

[39] B Jöricke, Envelopes of holomorphy and holomorphic discs, Invent. Math. 178 (2009) 73

[40] S Kaliman, M Zaĭdenberg, A tranversality theorem for holomorphic mappings and stability of Eisenman–Kobayashi measures, Trans. Amer. Math. Soc. 348 (1996) 661

[41] K Kodaira, D C Spencer, Divisor class groups on algebraic varieties, Proc. Nat. Acad. Sci. USA 39 (1953) 872

[42] W S Massey, Singular homology theory, 70, Springer (1980)

[43] J Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962) 337

[44] T Ohsawa, Completeness of noncompact analytic spaces, Publ. Res. Inst. Math. Sci. 20 (1984) 683

[45] M Peternell, q–completeness of subsets in complex projective space, Math. Z. 195 (1987) 443

[46] M Schneider, Über eine Vermutung von Hartshorne, Math. Ann. 201 (1973) 221

[47] G Sorani, Omologia degli spazi q–pseudoconvessi, Ann. Scuola Norm. Sup. Pisa 16 (1962) 299

[48] C Soulé, C Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198 (2005) 107

[49] E H Spanier, Algebraic topology, Springer (1981)

[50] B Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997) 467

[51] S Trivedi, Stratified transversality of holomorphic maps, Internat. J. Math. 24 (2013) 1350106, 12

[52] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994)