Let X be a compact, connected, algebraic n-dimensional submanifold of h% and let Y be a hyperplane in F r. There are two theorems of Lefschetz [14] that relate the homology ofX~Yto that of X.
First Lefschetz Theorem:Hi(X n Y, L r) ~ HI(X, ;E) is an isomorphism for i< n-1, and H~_ 1 (X n Y, 7/)---, H,_ t (X, 2~) is a surjection.(A) The Second Lefschetz Theorem gives an explicit description of ker, the kernel of H,_ l(Xn Y,~)~H,_ t(X,C) when X meets Y transversely, and expresses H,,_ l(Xn Y,¢)= ker@inv, where Q is with respect to the middle homology pairing. The subspace ker is irreducible under an appropriate monodromy action, and inv, the space of cycles invariant under this action, is also described.After work by Barth [3,4], Larsen [16], Hartshorne [13], and Ogus [20], a complete generalization of the first theorem was given by Sommese [21, 22]. The result of [4] that I use is : Let X, Ybe compact, connected algebraic submanifolds of lP~, of dimensions n and r-s, respectively. Assume that s+n