Abundance theorem for minimal compact K\"ahler manifolds with vanishing second Chern class

TL;DR

This paper investigates the abundance conjecture for minimal compact K"ahler manifolds with nef cotangent bundles, establishing conditions under which the second Chern class vanishes and the canonical bundle is semi-ample, advancing understanding in complex geometry.

Contribution

It characterizes when the second Chern class vanishes for such manifolds and proves semi-ampleness of the canonical bundle under these conditions, linking geometric properties.

Findings

Second Chern class vanishes iff cotangent bundle is nef and canonical bundle has numerical dimension 0 or 1

Canonical bundle is semi-ample in these cases

Relation established between fiber variation and cotangent bundle semipositivity

Abstract

In this paper, for compact K\"ahler manifolds with nef cotangent bundle, we study the abundance conjecture and the associated Iitaka fibrations. We show that, for a minimal compact K\"ahler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has the numerical dimension $0$ or $1$. Additionally, in this case, we prove that the canonical bundle is semi-ample. Furthermore, we give a relation between the variation of the fibers of the Iitaka fibration and a certain semipositivity of the cotangent bundle.