Nagoya Mathematical Journal

Morse inequalities for covering manifolds

Radu Todor, Ionuţ Chiose, and George Marinescu

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We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of $L^2$ 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.

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Nagoya Math. J., Volume 163 (2001), 145-165.

First available in Project Euclid: 27 April 2005

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Zentralblatt MATH identifier

Primary: 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30]
Secondary: 32F10: $q$-convexity, $q$-concavity 32Q55: Topological aspects of complex manifolds 58J37: Perturbations; asymptotics


Todor, Radu; Chiose, Ionuţ; Marinescu, George. Morse inequalities for covering manifolds. Nagoya Math. J. 163 (2001), 145--165.

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  • A. Andreotti, Théorèmes de dépendance algébrique sur les espaces,complexes pseudo-concaves , Bull. Soc. Math. France, 91 (1963), 1–38.
  • A. Andreotti, H. Grauert, Théorème de finitude pour la cohomologie des espaces complexes , Bull. Soc. Math. France, 90 (1962), 193–259.
  • A. Andreotti, E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds , Inst. Hautes Etudes Sci. Publ. Math., 25 (1965), 81–130.
  • A. Andreotti, G. Tomassini, Some remarks on pseudoconcave manifolds , Essays in Topology and Related Topics, dedicated to G. de Rham, Springer (1970), 84–105, Berlin–Heidelberg–New York (R. Narasimhan, A. Haefliger, eds.).
  • M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras , Astérisque (1976, 32–33 ), 43–72.
  • T. Bouche, Inégalités de Morse pour la $d''$–cohomologie sur une variété non– compacte , Ann. Sci. Ecole Norm. Sup., 22 (1989), 501–513.
  • C. Bănică, O. Stănăşilă, Algebraic methods in the global theory of complex spaces, Wiley, New York(1976).
  • H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger operators with applications to quantum physics, Springer–Verlag(1987, Texts and Monographs in Physics).
  • J. P. Demailly, Champs magnétiques et inégalités de Morse pour la $d''$–cohomologie , Ann. Inst. Fourier, 35 (1985), 189–229. \comment{
  • J. P. Demailly, Regularization of closed positive currents and intersection theory , J. Alg. Geom., 1(1992), 361–409.
  • H. Grauert, O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen , Invent. Math., 11 (1970), 263–292. }
  • Ph. A. Griffiths, The extension problem in complex analysis; embedding with positive normal bundle , Amer. J. Math., 88 (1966), 366–446.
  • A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North–Holland, Amsterdam(1968).
  • M. Gromov, M. G. Henkin, M. Shubin, $L^2$ holomorphic functions on pseudo–convex coverings , GAFA, 8 (1998), 552–585.
  • G.Henniart, Les inégalités de Morse (d'après Witten) , Astérisque, no. 121/122, 1983/84 , 43–61(1985).
  • J. Kollár, Shafarevich maps and automorphic forms, Princeton University Press, Princeton, NJ (1995).
  • L. Lempert, Embeddings of three dimensional Cauchy–Riemann manifolds , Math. Ann., 300, 1-15(1994).
  • G. Marinescu, Asymptotic Morse Inequalities for Pseudoconcave Manifolds , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996, no. 1), 27–55.
  • A. Nadel, H. Tsuji, Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom., 28 (1988, no. 3), 503–512.
  • A. Nadel, On complex manifolds which can be compactified by adding finitely many points , Invent. Math., 101 (1990, no. 1), 173–189. \comment{
  • T. Napier, Convexity properties of coverings of smooth projective varieties, Math. Ann., 286 (1990), 433–479.. }
  • T. Napier, M. Ramachandran, The $L^2$–method, weak Lefschetz theorems and the topology of Kähler manifolds , JAMS, 11, no. 2, 375–396.
  • M.V. Nori, Zariski's conjecture and related problems , Ann. Sci. Ec. Norm. Sup., 16(1983), 305–344. \comment{
  • T. Ohsawa, Hodge spectral sequence and symmetry on compact Kähler spaces, Publ. Res. Inst. Math. Sci., 23(1987), 613–625. }
  • T. Ohsawa, Isomorphism theorems for cohomology groups of weakly $1$–complete manifolds , Publ. Res. Inst. Math. Sci., 18(1982), 191–232.
  • H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave boundary , Proceedings of the Conference on Complex Analysis (Minneapolis 1964), Springer–Verlag, Berlin (1965), 242–256.
  • L. Saper, $L^2$-cohomology and intersection homology of certain algebraic varieties with isolated singularities , Invent. Math., 82 (1985, no. 2), 207–255.
  • Y. T. Siu, A vanishing theorem for semipositive line bundles over non-Kähler manifolds , J. Diff. Geom., 19 , 431–452 (1984). \comment{
  • Y. T. Siu, Some recent results in complex manifold theory related to vanishing theorems for the semipositive case , Lecture Notes in Math., 169–192, 1111 , Springer, Berlin-New York (1985, Arbeittagung Bonn 1984). }
  • Y. T. Siu, S. T. Yau, Compactification of negatively curved complete Kähler manifolds of finite volume , Ann. Math. Stud., 102, 363–380(1982).
  • M. Shubin, Semiclassical asymptotics on covering manifolds and Morse inequalities , GAFA, 6 (1996, no. 2), 370–409.
  • S. Takayama, A differential geometric property of big line bundles , Tôhoku Math. J. (2), 46 (1994, no. 2), 281–291. \comment{
  • S. Zucker, Hodge theory with degenerating coefficients: $L^2$ cohomology in the Poincaré metric , Ann. Math., 109, 415–476(1979). }
  • E. Witten, Supersymmetry and Morse theory , J. Diff. Geom., 17, 661–692 (1982).