On the weighted orthogonal Ricci curvature

TL;DR

This paper introduces a new two-parameter curvature concept called weighted orthogonal Ricci curvature, which helps analyze relationships between different curvature types in Kähler and Hermitian manifolds, leading to new vanishing theorems.

Contribution

It defines the weighted orthogonal Ricci curvature and demonstrates its usefulness in establishing curvature constraints and vanishing theorems for complex manifolds.

Findings

Introduces weighted orthogonal Ricci curvature as a new geometric tool.

Establishes vanishing theorems using this curvature in Kähler and Hermitian contexts.

Provides insights into curvature conditions for projective Kähler manifolds.

Abstract

We introduce the weighted orthogonal Ricci curvature -- a two-parameter version of Ni--Zheng's orthogonal Ricci curvature. This curvature serves as a very natural object in the study of the relationship between the Ricci curvature(s) and the holomorphic sectional curvature. In particular, in determining optimal curvature constraints for a compact K\"ahler manifold to be projective. In this direction, we prove a number of vanishing theorems using the weighted orthogonal Ricci curvature(s) in both the K\"ahler and Hermitian category.