A LEFSCHETZ THEOREM ON THE PICARD GROUP OF COMPLEX PROJECTIVE VARIETIES

Abstract

In this paper a theorem of Lefschetz type is a theorem which compares geometric invariants of a subvariety of the projective space and the ones of a hyperplane section of this subvariety. The prototype is the theorem of S. Lefschetz which states that the k th homology of a non-singular complex projective variety of dimension n equals the Ic th homology of a hyperplane section for Ic I n 2. This type of results was generalized to homotopy groups by R. Bott ([B]) using the Morse theory approach of A. Andreotti and T. Frankel ([AF]). In parallel with the Lefschetz theorem which compares the (algebraic) fundamental group of a non-singular projective algebraic variety of dimension 2 3 and the one of a hyperplane section of this variety, A. Grothendieck [G2] considered cases when a theorem of Lefschetz type for the Picard group could be obtained.