Reducible hyperplane sections I

Dedicated to the memory of our friend and colleague, Michael Schneider

抄録

In this article we begin the study of \hat{X}, an n-dimensional algebraic submanifold of complex projective space \bm{P}^N, in terms of a hyperplane section A which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of \hat{X} to the cohomology of the components of a normal crossing divisor which is ample, and a strong extension theorem for divisors which are high index Fano fibrations. As a consequence we describe \hat{X}=\bm{P}^N of dimension at least five if the intersection of \hat{X} with some hyperplane is a union of r≥q 2 smooth normal crossing divisors \hat{A₁}, ..., \hat{A_r}, such that for each i, h¹(\mathcal{O}_{\hat{A_i}}) equals the genus g(\hat{A_i}) of a curve section of \hat{A_i}. Complete results are also given for the case of dimension four when r=2.