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

Abstract

Let $X$ be a complex submanifold of dimension $d$ of $\mathbb P^m\times\mathbb P^n$ ($m\geq n\geq 2$) 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 $t:=\max\{\dimπ_1(X),\dimπ_2(X)\}$, where $π_1$ and $π_2$ 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 $d\geq \frac{m+n+t+1}{2}$ then $X$ is algebraically simply connected, the map $α$ is injective and $\Coker(α)$ is torsion-free. Moreover $α$ is an isomorphism if $d\geq\frac{m+n+t+2}{2}$, 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}).