On orthogonal Ricci curvature
On orthogonal Ricci curvature
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关于正交里奇曲率
DOI：
10.1090/conm/735/14827
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
F. Zheng
中科院分区：
文献类型：
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作者：
Lei Ni;F. Zheng
本文讨论了具有正正交Ricci曲率的紧致Kähler流形的一些最新研究进展。正正交Ricci曲率是一个曲率条件，定义为Ricci曲率与全纯截面曲率之差。在作者的近期工作以及作者与Q. Wang的比较定理、消失定理和结构定理已被证明。我们构造了这类流形的例子，并给出了低维流形的分类结果。1.设（M，g）是复维数n的Kähler流形.它的正交Ricci曲率Ric由下式定义：[21])Ric = Ric（X，X）−R（X，X，X，X）/|X|其中X是在点x ∈ M处的非零型（1，0）切向量.这种曲率出现在凯勒流形的比较定理的研究和以前的研究流形与所谓的非负二次正交平分曲率（cf. [4]，[26]，[16]，[5]）。我们建议读者参考[21]，以获得关于这个主题的更详细的说明。显然，这个曲率与Ricci曲率Ric和全纯截面曲率H密切相关。很自然地，我们会问，Ric和Ric或H之间的关系是什么（除了对单位长度切向量Ric + H = Ric的明显关系之外），以及什么样的紧致复流形M可以在任何地方都允许Ric> 0（或≥ 0，或≤ 0，或< 0，或<0）的Kähler度量？在这篇文章中，我们将集中讨论曲率条件Ric_∞，并特别注意处处Ric_∞> 0的紧Kähler流形类，除了在第2节中，也考虑了完备非紧Kähler流形。在本文中，除非另有说明，我们将假设复维数n ≥ 2，因为当n = 1时Ric ≥ 0。我们从以下观察开始。在一点x ∈ M，让我们用S2 n −1 x表示在x处的所有（1，0）型切向量的单位球面。根据Berger的经典结果，LN的研究得到了NSF基金DMS-1401500和“科技创新能力建设-基础研究基金”的部分支持。FZ的研究部分得到了西蒙斯合作基金355557的支持。
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.