Comparison and vanishing theorems for Kähler manifolds

Abstract

In this paper, we consider orthogonal Ricci curvature Ric ⊥ for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate their subtle relationship. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. The simply-connectedness is also shown when the complex dimension is smaller than five.

General Schwarz Lemmata and their applications

“…It was also shown in [20] via explicit examples that B ⊥ is independent of the holomorphic sectional curvature H, as well as the Ricci curvature. Similarly Ric ⊥ is independent of Ric, as well as H. It was proved in [20] that for manifold whose Ric ⊥ has a positive lower bound, the manifold is compact with an effective diameter uppper bound. (See [25] for the corresponding result for holomorphic sectional curvature.)…”

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

“…Note that in this case the bisectional curvature lower bound can be easily checked via the polarization formula (cf. [32], formula in the proof of corollary 2.1), and the result can be stated locally given that the global result is derived from a local estimate. This result can be viewed as a stability statement of the classical result asserting that a complete Kähler manifold with the negative constant holomorphic sectional curvature must be a quotient of the complex hyperbolic space form.…”

Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature

“…Hence understanding the structure of Kähler manifolds with NOB is an interesting area of research. In view of the examples constructed in [18], [26] and [30] the orthogonal bisectional curvature B ⊥ is completely independent of the holomorphic sectional curvature, or the Ricci curvature. Recently there is a joint work of second author [26] proving a Liouville theorem for plurisubharmonic functions, which complements a recent result of Liu [20], and a gap theorem in on Kähler manifolds with B ⊥ ≥ 0 and Ric ≥ 0.…”

“…Clearly one can not expect such a result for general Kähler manifolds. For example, in [30] a complete unitary symmetric metric was constructed on C n with B ⊥ > 0 and Ric > 0. Apart from the motivation from the study of complex structure of Kähler manifolds with B ⊥ ≥ 0, the above result is motivated by a recent work of Munteanu-Wang [21], where a similar statement was proved for gradient shrinking Ricci soltions under the assumption that the sectional curvature is nonnegative and Ricci is positive.…”