Inequalities of Miyaoka-Yau type $\&$ Uniformisation of varieties of intermediate Kodaira Dimension

TL;DR

This paper establishes new inequalities relating Chern classes of minimal complex projective varieties across all dimensions and numerical dimensions, resolving the Abundance conjecture in certain cases and characterizing extremal varieties.

Contribution

It introduces a comprehensive set of inequalities for Chern classes of varieties of intermediate Kodaira dimension, extending known results and providing sharpness and equality characterizations.

Findings

Derived inequalities for Chern classes valid for all dimensions and numerical dimensions.

Resolved the Abundance conjecture in specific cases where inequalities are sharp.

Provided new examples of extremal varieties with optimal Chern class configurations.

Abstract

In this paper we present, for any integers $0\leq \nu \leq n$, a set of inequalities satisfied by the Chern classes of any minimal complex projective variety of dimension $n$ and numerical dimension $\nu$. In the cases where $\nu$ is either very small or very large compared with $n$, this recovers many previously known results. We demonstrate that our inequalities are sharp by providing an explicit characterisation of those varieties achieving the equality; our proof, in particular, resolves the Abundance conjecture in this situation. Additionally, we provide some new examples of varieties with extremal Chern classes that demonstrate the optimality of our results.