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

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dc.creatorBadescu, Lucian
dc.creatorRepetto, Flavia
dc.date2007-01-13
dc.date.accessioned2026-07-07T07:40:59Z
dc.date.available2026-07-07T07:40:59Z
dc.descriptionLet $X$ 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}).
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/math/0701376
dc.identifierhttp://arxiv.org/abs/math/0701376
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/121949
dc.subjectAlgebraic Geometry
dc.subject14M07; 14B20
dc.titleA Barth-Lefschetz theorem for submanifolds of a product of projective spaces
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