On orthogonal Ricci curvature

In this paper we discuss some recent progresses in the study of compact Kähler manifolds with positive orthogonal Ricci curvature, a curvature condition defined as the difference between Ricci curvature and holomorphic sectional curvature. In the recent works by authors and the joint work of authors with Q. Wang the comparison theorems, vanishing theorems, and structural theorems for such manifolds have been proved. We also constructed examples of this type of manifolds, and give some classification results in low dimensions. 1. Orthogonal Ricci curvature Let (M, g) be a Kähler manifold of complex dimension n. Its orthogonal Ricci curvature Ric⊥ is defined by (cf. [21]): Ric⊥ XX = Ric(X,X)−R(X,X,X,X)/|X|, where X is a non-zero type (1, 0) tangent vector at a point x ∈ M. This curvature arises in the study of the comparison theorem for Kähler manifolds and the previous study of manifolds with so-called nonnegative quadratic orthogonal bisectional curvature (cf. [4], [26], [16], [5]). We refer the readers to [21] for a more detailed account on this topic. Clearly this curvature is closely related to Ricci curvature Ric and holomorphic sectional curvature H. It is natural to ask, what is the relationship between Ric⊥ and Ric or H (other than the obvious one that Ric⊥ + H = Ric for unit length tangent vectors), and what kind of compact complex manifolds M can admit Kähler metrics with Ric⊥ > 0 (or ≥ 0, or ≤ 0, or < 0, or ≡ 0) everywhere? In this paper, we will focus on the curvature condition Ric⊥ and pay particular attention to the class of compact Kähler manifolds with Ric⊥ > 0 everywhere, except in Section 2 where complete noncompact Kähler manifolds are also considered. Throughout this paper, we will assume that the complex dimension n ≥ 2 unless stated otherwise, since Ric⊥ ≡ 0 when n = 1. We start with the following observation. At a point x ∈ M, let us denote by S2n−1 x the unit sphere of all type (1, 0) tangent vector at x of unit length. By a classic result of Berger, The research of LN is partially supported by NSF grant DMS-1401500 and the “Capacity Building for Sci-Tech Innovation-Fundamental Research Funds”. The research of FZ is partially supported by a Simons Collaboration Grant 355557.