A GENERALIZATION OF LEFSCHETZ-ZARISKI THEOREM ON FUNDAMENTAL GROUPS OF ALGEBRAIC VARIETIES

Let X and Y be the complements of divisors on non-singular irreducible closed subvarieties [Formula: see text] and [Formula: see text] in ℙn, respectively. Suppose that dim X+ dim Y≥n+2. Then, for a general g∈PGL (n+1), the natural homomorphism π1(g(X)∩Y)→π1(Y) induces a surjection from Ker (π1(g(X)∩Y)→π1(g(X))) onto π1(Y), and there is a surjection to its kernel from the cokernel of π2(X)→π2(ℙn). In particular, if E⊂ℙn is a hypersurface and 2·dim [Formula: see text], then [Formula: see text] is isomorphic to π1(ℙn\E) for a general g∈PGL (n+1).