Topological Properties of Cyclic Coverings Branched Along an Ample Divisor

Abstract

Let X’ → X be a finite morphism between two complex connected projective k-folds. Since Π is surjective, the Betti numbers of X and X’ are related as follows(0.1) bi(X) ≦ bi(X’).In particular, if Π is a cyclic covering and the branch locus A is an ample divisor, (0.1) is in fact an equality for i ≦ k — 1 (see 1.10 or, more generally, [5] ). It seems natural to look for such coverings satisfying(0.2) bk(X)= bk(X’).Let us see what happens for k = 2. In this case (0.2) can be rephrased as(0.3) 2x(Ox) + h1,1 (X) + g(Δ) = 2,