A Barth-Lefschetz theorem for submanifolds of a product of projective spaces

Abstract

Let $X$ be a complex submanifold of dimension $d$ of ${\mathbb{P}}^m\times {\mathbb{P}}^n$ ($m\ge n\ge 2$) and denote by $α\colon\Pic(\mathbb P^m\times\mathbb P^n)\to \Pic(X)$ the restriction map of Picard groups, by $N_{X|{\mathbb{P}}^m\times {\mathbb{P}}^n}$ the normal bundle of $X$ in ${\mathbb{P}}^m\times {\mathbb{P}}^n$. Set $t:=\max \{\dim {\pi }_1(X),\dim {\pi }_2(X)\}$, where ${\pi }_1$ and ${\pi }_2$ are the two projections of ${\mathbb{P}}^m\times {\mathbb{P}}^n$. We prove a Barth-Lefschetz type result as follows: {\em Theorem.} {\it If $d\ge \frac{m+n+t+1}{2}$ then $X$ is algebraically simply connected, the map $\alpha$ is injective and $\Coker(α)$ is torsion-free. Moreover $\alpha$ is an isomorphism if $d\ge \frac{m+n+t+2}{2}$, or if $d=\frac{m+n+t+1}{2}$ and $N_{X|{\mathbb{P}}^m\times {\mathbb{P}}^n}$ is decomposable.} These bounds are optimal. The main technical ingredients in the proof are: the Kodaira-Le Potier vanishing theorem in the generalized form of Sommese (\cite{LP}, \cite{ShS}), the join construction and an algebraisation result of Faltings concerning small codimensional subvarieties in ${\mathbb{P}}^N$ (see \cite{Fa}).

18 pages