On the weighted orthogonal Ricci curvature

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 Kähler manifold to be projective. In this direction, we prove a number of vanishing theorems using the weighted orthogonal Ricci curvature(s) in both the Kähler and Hermitian category.

On the altered holomorphic curvatures of Hermitian manifolds

“…Let x 0 ∈ M be the point at which the comass σ 0 attains its maximum. From Ni's viscosity considerations [40] (c.f., [41,14]), we have (in a fixed unitary frame e k near x 0 )…”

Remarks on the quadratic orthogonal bisectional curvature

“…In contrast with the orthogonal bisectional curvature, the QOBC is strictly weaker than the bisectional curvature, with an explicit example constructed in [18]. The QOBC has been the subject of large interest in recent years (see, e.g., [6,7,8,9,10,11,17,18,21,22,29]). The purpose of the present short note is to describe the link between combinatorics, distance geometry, and the QOBC.…”