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The $L^2$ $\bar \partial$-method, weak Lefschetz theorems, and the topology of Kähler manifolds

Authors: Terrence Napier and Mohan Ramachandran
Journal: J. Amer. Math. Soc. 11 (1998), 375-396
MSC (1991): Primary 14E20, 32C10, 32C17
MathSciNet review: 1477601
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Abstract: A new approach to Nori’s weak Lefschetz theorem is described. The new approach, which involves the $\bar \partial$-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if $X$ and $Y$ are connected smooth projective varieties of positive dimension and $f : Y \rightarrow X$ is a holomorphic immersion with ample normal bundle, then the image of $\pi _1(Y)$ in $\pi _1(X)$ is of finite index. This result is obtained as a consequence of a direct generalization of Nori’s theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in $\Bbb C ^n$ ($n\geq 3$) by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold $X$ of dimension $n\geq 3$ has a strongly pseudoconvex end $E$ and $\text {Ricci} (X) \leq -C$ for some positive constant $C$, then, away from $E$, $X$ has finite volume.

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  • Aldo Andreotti, Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France 91 (1963), 1–38 (French). MR 152674
  • Aldo Andreotti and Hans Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259 (French). MR 150342
  • Aldo Andreotti and Edoardo Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81–130. MR 175148
  • Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981
  • Constantin Bănică and Octavian Stănăşilă, Algebraic methods in the global theory of complex spaces, Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976. Translated from the Romanian. MR 0463470
  • D. M. Burns, in preparation.
  • Frédéric Campana, Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France 122 (1994), no. 2, 255–284 (French, with English and French summaries). MR 1273904
  • F. Campana, Negativity of compact curves in infinite covers of projective surfaces, preprint.
  • Jean-Pierre Demailly, Estimations $L^{2}$ pour l’opérateur $\bar \partial $ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 457–511 (French). MR 690650
  • Jean-Pierre Demailly, Champs magnétiques et inégalités de Morse pour la $d”$-cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 189–229 (French, with English summary). MR 812325
  • William Fulton, On the topology of algebraic varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 15–46. MR 927947
  • William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
  • William Fulton and Robert Lazarsfeld, Positivity and excess intersection, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 97–105. MR 685765
  • M. Gaffney, A special Stokes theorem for Riemannian manifolds, Ann. of Math. 60 (1954), 140–145.
  • Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, DOI
  • Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
  • Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
  • Heisuke Hironaka and Hideyuki Matsumura, Formal functions and formal embeddings, J. Math. Soc. Japan 20 (1968), 52–82. MR 251043, DOI
  • Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematical Library, Vol. 7. MR 0344507
  • János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589
  • László Lempert, Algebraic approximations in analytic geometry, Invent. Math. 121 (1995), no. 2, 335–353. MR 1346210, DOI
  • Alan Nadel and Hajime Tsuji, Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom. 28 (1988), no. 3, 503–512. MR 965227
  • Shigeo Nakano, Vanishing theorems for weakly 1-complete manifolds. II, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 101–110. MR 0382735, DOI
  • T. Napier and M. Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5 (1995), no. 5, 809–851. MR 1354291, DOI
  • Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347
  • Christian Okonek, Concavity, convexity and complements in complex spaces, Complex analysis and algebraic geometry (Göttingen, 1985) Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 104–126. MR 855879, DOI
  • H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 242–256. MR 0176106
  • F. Sakai, Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 239–257. MR 0590433
  • Yum Tong Siu and Shing Tung Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 363–380. MR 645748
  • Henri Skoda, Morphismes surjectifs et fibrés linéaires semi-positifs, Séminaire Pierre Lelong-Henri Skoda (Analyse), Année 1976/77, Lecture Notes in Math., vol. 694, Springer, Berlin, 1978, pp. 290–324 (French). MR 522481
  • Andrew John Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), no. 3, 229–256. MR 466647, DOI
  • Shigeharu Takayama, A differential geometric property of big line bundles, Tohoku Math. J. (2) 46 (1994), no. 2, 281–291. MR 1272883, DOI

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Additional Information

Terrence Napier
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015

Mohan Ramachandran
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214

Keywords: Fundamental group, projective variety, line bundle, ball quotient
Received by editor(s): July 8, 1997
Received by editor(s) in revised form: November 4, 1997
Additional Notes: The authors’ research was partially supported by NSF grants DMS9411154 (T.N.) and DMS9626169 (M.R.).
Article copyright: © Copyright 1998 American Mathematical Society