Chern numbers on positive vector bundles and combinatorics

TL;DR

This paper develops combinatorial methods to study Chern numbers on positive vector bundles, establishing bounds and ordering properties, and explores positivity conjectures for complex manifolds.

Contribution

It introduces combinatorial techniques to derive bounds and orderings of Chern numbers, advancing understanding of positivity for vector bundles and complex manifolds.

Findings

Established a lower bound for Chern numbers of ample vector bundles

Proved Chern numbers on nef vector bundles obey reverse dominance ordering

Confirmed positivity conjectures for compact homogeneous complex manifolds

Abstract

Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress towards a long-standing question. Along this line we prove that Chern numbers on nef vector bundles obey reverse dominance ordering, which improves upon some classical and recent results. We propose a simultaneous positivity question on (signed) Chern numbers of compact complex or K\"{a}hler manifolds whose (co)tangent bundles are semipositive in various senses, and show that it holds true for compact homogeneous complex manifolds.