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

Abstract

Let X be a complex submanifold of dimension d of ℙm × ℙn (m ≥ n ≥ 2) and denote by α: Pic(ℙm × ℙn) → Pic(X) the restriction map of Picard groups, by NX|ℙm × ℙn the normal bundle of X in ℙm × ℙn. Set t := max{dim π1(X), dim π2(X)}, where π1 and π2 are the two projections of ℙm × ℙn. We prove a Barth–Lefschetz type result as follows: Theorem. If [Formula: see text] then X is algebraically simply connected, the map α is injective and Coker(α) is torsion-free. Moreover α is an isomorphism if [Formula: see text], or if [Formula: see text] and NX|ℙm×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 ℙN (see [9]).