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D G ] 2 7 O ct 2 01 7 COMPACT KÄHLER MANIFOLDS WITH POSITIVE ORTHOGONAL BISECTIONAL CURVATURE
| Content Provider | Semantic Scholar |
|---|---|
| Author | Feng, Huitao Liu, Kefeng Wan, Xueyuan |
| Copyright Year | 2018 |
| Abstract | In this short note, using Siu-Yau’s method [14], we give a new proof that any n-dimensional compact Kähler manifold with positive orthogonal bisectional curvature must be biholomorphic to P. 0. Introduction In the celebrated paper [14], Siu and Yau presented a differential geometric proof of the famous Frankel conjecture in Kähler geometry, which states that a compact Kähler manifold M with positive bisectional curvature must be biholomorphic to P. Using the method of Siu-Yau, Seaman [12] in 1993 proved that any compact Kähler manifold M with positive curvature on totally isotropic 2-planes must be biholomorphic to P. On the other hand, there is also a concept of orthogonal bisectional curvature (cf. Definition 1.1 in this paper) for Kähler manifolds of dimension n ≥ 2, which was introduced by Cao and Hamilton in the late 1980’s. Note that the condition of positive orthogonal bisectional curvature is weaker than both the conditions of positive bisectional curvature and positive curvature on totally isotropic 2-planes for a Kähler manifold M with n ≥ 2. Hence a natural question is whether a compact Kähler manifold M (n ≥ 2) with positive orthogonal bisectional curvature is biholomorphic to P. Inspired by an observation of Cao and Hamilton in an unpublished work that the nonnegativity of the orthogonal bisectional curvature is preserved under the Kähler-Ricci flow, Chen [3] in 2007 gave a positive answer, under the extra condition c1(M) > 0, to this question by using the Kähler-Ricci flow method. Chen further asked whether this extra condition could be dropped. Later, H. Gu and Z. Zhang [7] in 2010 showed that c1(M) > 0 holds for any compact Kähler manifold M with positive orthogonal bisectional curvature. Therefore, by combining the results of Chen and Gu-Zhang, one has Theorem 0.1. Let M be an n-dimensional compact Kähler manifold with positive orthogonal bisectional curvature, then M is biholomorphic to P. In this note, we will give a new proof of the above theorem by using Siu-Yau’s method [14], which answers a question of Chen raised in [3]. Note that Chen’s proof depends heavily on the Kähler-Ricci flow techniques. More precisely, by assuming c1(M) > 0, he can show that a Kähler metric with positive orthogonal bisectional curvature flows to a metric of positive 1 Partially supported by NSFC (Grant No. 11221091, 11271062, 11571184). 3 Partially supported by NSFC (Grant No. 11221091,11571184) and the Ph.D. Candidate Research Innovation Fund of Nankai University. 1Their proof has since appeared, see Theorem 2.3 of [2]. 1 2 Huitao Feng, Kefeng Liu and Xueyuan Wan holomorphic bisectional curvature, and then he got a proof of Theorem 0.1 by using Siu-Yau’s result. Compared with Chen’s proof, our proof is more direct and geometric. This note is organized as follows. In Section 1, by computing directly the second variation of the energy of maps f : P1 → (M,h), we prove that an energy minimal map f must be holomorphic or conjugate holomorphic when (M,h) is a compact Kähler manifold with positive orthogonal bisectional curvature, which is the key step of our proof of Theorem 0.1. In Section 2, we will complete the proof of Theorem 0.1 by using Siu-Yau’s method [14]. Acknowledgements. The authors would like to thank Professor Xiaokui Yang for his helpful suggestions in preparing this paper. The authors would like to thank the reviewers for their comments that help improve the paper. 1. Complex Analyticity of Energy Minimizing Maps In this section, we first compute the first and second variations of the total energy of smooth maps f : P1 → (M,h), and then prove the main result of this section that an energy minimal map f must be holomorphic or conjugate holomorphic when (M,h) is a compact Kähler manifold with positive orthogonal bisectional curvature. Note that Siu-Yau [14] used ∂̄-energy in their proof. Let P1 be the complex projective space of complex dimension one with a fixed conformal structure ω and M a compact Kähler manifold with a Kähler metric h. In local holomorphic coordinates on P1 and M , we write ω and h as following respectively: ω = λdw ⊗ dw̄, h = n |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://export.arxiv.org/pdf/1710.10240 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
