A Barth–lefschetz Theorem for Submanifolds of a Product of Projective Spaces

Abstract

Let X be a complex submanifold of dimension d of P m ×P n (m ≥ n ≥ 2) and denote by α : Pic(P m × P n ) → Pic(X) the restriction map of Picard groups, by N X|P m ×P n the normal bundle of X in P m × P n . Set t := max{dim π 1 (X), dim π 2 (X)}, where π 1 and π 2 are the two projections of P m × P n . We prove a Barth-Lefschetz type result as follows: Theorem. 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 N X|P m ×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 ([18], [19]), the join construction and an algebraisation result of Faltings concerning small codimensional subvarieties in P N (see [9]).