The 𝐿² ∂̄-method, weak Lefschetz theorems, and the topology of Kähler manifolds

Napier, Terrence¹ ; Ramachandran, Mohan²

¹ Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
² Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14214

Journal of the American Mathematical Society, Tome 11 (1998) no. 2, pp. 375-396

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-98-00257-4

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