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Morse inequalities for covering manifolds

Published online by Cambridge University Press:  22 January 2016

Radu Todor ,
Ionuţ Chiose  and
George Marinescu
Radu Todor
Affiliation:
Faculty of Mathematics, University of Bucharest, Str Academiei 14, Bucharest, Romania
Ionuţ Chiose
Affiliation:
Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USAchiose@math.sunysb.edu
George Marinescu*
Affiliation:
Institute of Mathematics of the Romanian Academy, Bucharest, Romania
*
Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Deutschlandgeorge@mathematik.hu-berlin.de
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Abstract

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We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

Keywords

58J3732F1032Q55
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 163 , September 2001 , pp. 145 - 165
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

Footnotes

1

Supported by a DFG Stipendium at the Graduiertenkolleg ‘Geometrie und Nichtlineare Analysis’ at Humboldt-Universität zu Berlin and SFB 288.

References

[1] Andreotti, A., Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France, 91 (1963), 1–38.Google Scholar
[2] Andreotti, A., Grauert, H., Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193–259.Google Scholar
[3] Andreotti, A., Vesentini, E., Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Etudes Sci. Publ. Math., 25 (1965), 81–130.Google Scholar
[4] Andreotti, A., Tomassini, G., Some remarks on pseudoconcave manifolds, Essays in Topology and Related Topics, dedicated to G. de Rham (Narasimhan, R., Hae-fliger, A., eds.), Springer, Berlin–Heidelberg–New York (1970), pp. 84–105.Google Scholar
[5] Atiyah, M. F., Elliptic operators, discrete groups and von Neumann algebras, Astérisque, 32–33 (1976), pp. 43–72.Google Scholar
[6] Bouche, T., Inégalités de Morse pour la d″-cohomologie sur une variété noncompacte, Ann. Sci. Ecole Norm. Sup., 22 (1989), 501–513.Google Scholar
[7] Bănică, C., Stănăşilă, O., Algebraic methods in the global theory of complex spaces, Wiley, New York, 1976.Google Scholar
[8] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B., Schrödinger operators with applications to quantum physics, Texts and Monographs in Physics, Springer-Verlag, 1987.Google Scholar
[9] Demailly, J. P., Champs magnétiques et inégalités de Morse pour la d″-cohomologie, Ann. Inst. Fourier, 35 (1985), 189–229.Google Scholar
[10] Griffiths, Ph. A., The extension problem in complex analysis; embedding with positive normal bundle, Amer. J. Math., 88 (1966), 366–446.Google Scholar
[11] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North-Holland, Amsterdam, 1968.Google Scholar
[12] Gromov, M., Henkin, M. G., Shubin, M., L2 holomorphic functions on pseudo-convex coverings, GAFA, 8 (1998), 552–585.Google Scholar
[13] Henniart, G., Les inégalités de Morse (d’après Witten), Astérisque, 1983/84 (1985), no. 121/122, 43–61.Google Scholar
[14] Kollár, J., Shafarevich maps and automorphic forms, Princeton University Press, Princeton, NJ, 1995.Google Scholar
[15] Lempert, L., Embeddings of three dimensional Cauchy-Riemann manifolds, Math. Ann., 300 (1994), 1–15.Google Scholar
[16] Marinescu, G., Asymptotic Morse Inequalities for Pseudoconcave Manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), no. 1, 27–55.Google Scholar
[17] Nadel, A., Tsuji, H., Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom., 28 (1988), no. 3, 503–512.CrossRefGoogle Scholar
[18] Nadel, A., On complex manifolds which can be compactified by adding finitely many points, Invent. Math., 101 (1990), no. 1, 173–189.CrossRefGoogle Scholar
[19] Napier, T., Ramachandran, M., The L2-method, weak Lefschetz theorems and the topology of Kähler manifolds, JAMS, 11, no. 2, 375–396.Google Scholar
[20] Nori, M.V., Zariski’s conjecture and related problems, Ann. Sci. Ec. Norm. Sup., 16 (1983), 305–344.Google Scholar
[21] Ohsawa, T., Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci., 18 (1982), 191–232.CrossRefGoogle Scholar
[22] Rossi, H., Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proceedings of the Conference on Complex Analysis (Minneapolis 1964), Springer-Verlag, Berlin (1965), pp. 242–256.Google Scholar
[23] Saper, L., L2-cohomology and intersection homology of certain algebraic varieties with isolated singularities , Invent. Math., 82 (1985), no. 2, 207–255.Google Scholar
[24] Siu, Y. T., A vanishing theorem for semipositive line bundles over non-Kähler manifolds, J. Diff. Geom., 19 (1984), 431–452.Google Scholar
[25] Siu, Y. T., Yau, S. T., Compactification of negatively curved complete Kähler manifolds of finite volume, Ann. Math. Stud., 102 (1982), 363–380.Google Scholar
[26] Shubin, M., Semiclassical asymptotics on covering manifolds and Morse inequalities, GAFA, 6 (1996), no. 2, 370–409.Google Scholar
[27] Takayama, S., A differential geometric property of big line bundles, Tôhoku Math. J. (2), 46 (1994), no. 2, 281–291.CrossRefGoogle Scholar
[28] Witten, E., Supersymmetry and Morse theory, J. Diff. Geom., 17 (1982), 661–692.Google Scholar