Inequalities of Miyaoka-Yau type $\&$ Uniformisation of varieties of intermediate Kodaira Dimension
Niklas Müller
TL;DR
The article develops Miyaoka–Yau-type Chern class inequalities for minimal projective varieties across intermediate numerical dimensions, providing a nonnegativity result for the polynomial $P_X(\varepsilon)$. It connects equality cases to a uniformisation framework in which $X$ becomes birational to a product by an Abelian variety modulo a finite group, with a ball quotient factor; the canonical divisor is shown to be semiample under these conditions. The authors develop a robust Higgs-sheaf toolkit on klt spaces, including twisted semistability, adapted differential sheaves, and Simpson correspondence, to derive higher-order inequalities and deduce semiampleness and uniformisation results. They also construct a counterexample demonstrating limitations of the converse implications and clarify the role of the Iitaka fibration in the uniformisation picture. Overall, the work extends classical Chern-number inequalities to varieties with intermediate Kodaira dimension and provides structural characterisations tied to ball quotients and abelian fibrations, deepening the link between Chern class data and global geometric structure.
Abstract
In this paper we present, for any integers $0\leq ν\leq n$, a set of inequalities satisfied by the Chern classes of any minimal complex projective variety of dimension $n$ and numerical dimension $ν$. In the cases where $ν$ 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.