2007/01/13 by Lucian Bădescu, Lucian Badescu, Badescu, Lucian +2
Computer Science · Mathematics · #14B20 #14M07 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.AG #msc:14B20 #msc:14M07
paper · pdf · doi:10.48550/arxiv.math/0701376
18 pages
arxiv created 2007/01/13 · openalex publication_date 2007/01/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a complex submanifold of dimension d of \mathbb Pm×\mathbb Pn (m≥ n≥ 2) and denote by α\colon\Pic(\mathbb Pm×\mathbb Pn)→ \Pic(X) the restriction map of Picard groups, by NX|\mathbb Pm×\mathbb Pn the normal bundle of X in \mathbb Pm×\mathbb Pn. Set t:=max\dimπ1(X),dimπ2(X)\, where π1 and π2 are the two projections of \mathbb Pm×\mathbb Pn. We prove a Barth-Lefschetz type result as follows: \em Theorem. \it If d≥ (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≥(m+n+t+2)/(2), or if d=(m+n+t+1)/(2) and NX|\mathbb Pm×\mathbb Pn 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 (\citeLP, \citeShS), the join construction and an algebraisation result of Faltings concerning small codimensional subvarieties in \mathbb PN (see \citeFa).