A BARTH-LEFSCHETZ THEOREM FOR SUBMANIFOLDS OF A PRODUCT OF PROJECTIVE SPACES

Let X be a complex submanifold of dimension d of P-m x P-n (m >= n >= 2) and denote by alpha: Pic(P-m x P-n) -> Pic(X) the restriction map of Picard groups, by N-X vertical bar Pm x Pn the normal bundle of X in P-m x P-n. Set t := max{dim pi(1)(X), dim pi(2)(X)}, where pi(1) and pi(2) are the two projections of P-m x 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 alpha is injective and Coker(alpha) is torsion-free. Moreover alpha is an isomorphism if d >= m+n+t+2/2, or if d = m+n+t+1/2 and N-X vertical bar Pm x 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 ([18, 19]), the join construction and an algebraization result of Faltings concerning small codimensional subvarieties in P-N (see [9]).