A view on extending morphisms from ample divisors

TL;DR

This paper surveys the extension of morphisms from ample divisors in projective manifolds, discussing classical principles, Lefschetz results, vanishing theorems, and modern techniques like Mori theory.

Contribution

It provides a comprehensive overview of problems, results, and conjectures related to extending morphisms from ample divisors using modern algebraic geometry tools.

Findings

Lefschetz type results facilitate understanding of morphism extensions.

Vanishing theorems play a key role in the theory.

Modern techniques like Mori theory offer new insights.

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.