Extension of log pluricanonical forms from subvarieties

TL;DR

This paper proves a general extension theorem for log pluricanonical systems and applies it to establish an optimal subadjunction theorem, advancing the understanding of canonical bundle positivity and the abundance conjecture.

Contribution

It introduces a broad extension theorem for log pluricanonical forms and derives an optimal subadjunction theorem linking positivity properties of canonical divisors.

Findings

Established a general extension theorem for log pluricanonical systems.

Derived an optimal subadjunction theorem relating canonical bundle positivity.

Provided an approach to the abundance conjecture via induction in dimension.

Abstract

In this paper, I prove a very general extension theorem for log pluricanonical systems. The main application of this extension theorem is (together with Kawamata's subadjunction theorem) to give an optimal subadjunction theorem which relates the positivities of canonical bundle of the ambient projective manifold and that of the (maximal) center of log canonical singularities. This is an extension of the corresponding result in my previous work where I dealt with log pluricanonical systems of general type. This subadjunction theorem indicates an approach to solve the abundance conjecture for canonical divisors (or log canonical divisors) in terms of the induction in dimension.