Rational curves on compact K\"ahler manifolds

TL;DR

This paper introduces an inductive approach to prove the existence of rational curves on certain compact Kähler manifolds with pseudoeffective canonical bundles, utilizing a new subadjunction formula based on positivity arguments.

Contribution

It develops a novel inductive strategy and a weak subadjunction formula for lc centers in the Kähler setting, extending techniques from projective geometry.

Findings

Establishes existence of rational curves on non-minimal compact Kähler manifolds with pseudoeffective canonical bundles.

Introduces a new subadjunction formula for lc centers in the Kähler context.

Adapts positivity arguments for relative adjoint classes to the Kähler setting.

Abstract

We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction formula for lc centers associated to certain big cohomology classes. This subadjunction formula is based, as in the projective case, on positivity arguments for relative adjoint classes.