The equality case in Wu-Yau inequalities

TL;DR

This paper investigates the conditions under which the inequalities used to establish the positivity of the canonical bundle in Kähler manifolds with quasi-negative curvature become equalities, providing insights into the geometric structure.

Contribution

It analyzes the equality cases of Wu-Yau type inequalities, offering new understanding of the geometric implications when these inequalities are tight.

Findings

Identifies conditions for equality in Wu-Yau inequalities.

Provides geometric interpretations of equality cases.

Enhances understanding of canonical bundle positivity in Kähler geometry.

Abstract

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact K\"ahler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.