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

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.