A view on extending morphisms from ample divisors

Abstract

The philosophy that ``a projective manifold is more special than any of its smooth hyperplane sections" was one of the classical principles of projective geometry. Lefschetz type results and related vanishing theorems were among the typically used techniques. We shall survey most of the problems, results and conjectures in this area, using the modern setting of ample divisors, and (some aspects of) Mori theory.Comment: To Appear in: Interactions of Classical and Numerical Algebraic Geometry, ed. by A. Bates, G. Besana and S. Di Rocco, Contemporary Mathematics, American Mathematical Societ