Reducible hyperplane sections I

In this article we begin the study of \hat{X} , an n -dimensional algebraic submanifold of complex projective space P^{N} , in terms of a hyperplane section \hat{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}\subset P^{N} of dimension at least five if the intersection of \hat{X} with some hyperplane is a union of r\geq 2 smooth normal crossing divisors A_{1} , . . . , A_{r} , such that for each i, h^{1}(\mathscr{O}_{\hat{\mathrm{A}}_{i}}) equals the genus g(A_{i}) of a curve section of A_{i} . Complete results are also given for the case of dimension four when r=2 .