Injectivity theorems with multiplier ideal sheaves and their applications

TL;DR

This survey discusses analytic injectivity theorems using L^2 methods and harmonic integrals, highlighting their applications in vanishing theorems, extension of sections, and semi-ampleness in birational geometry.

Contribution

It presents new analytic injectivity theorems based on L^2 methods and harmonic integrals, with applications to vanishing theorems and the minimal model program.

Findings

Derived Nadel type vanishing theorems

Extended sections of pluri-canonical sheaves

Results on semi-ampleness and the abundance conjecture

Abstract

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs. Moreover, we give some results on semi-ampleness related to the abundance conjecture in birational geometry (the minimal model program).