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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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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

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