Obstructions to the existence of Kähler structures on compact complex manifolds
Obstructions to the existence of Kähler structures on compact complex manifolds
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紧复流形上卡勒结构存在的障碍
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发表时间：
2014
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通讯作者：
Ionuţ Chiose
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作者：
Ionuţ Chiose
本文证明了Fujiki类C中的一个流形如果支持一个i-闭度量，则它是Kähler流形.这个结果意味着在Fujiki类C中的非Kähler的紧致复流形上，存在非零的二维（1，1）的正正流。介绍在[HaLa]，哈维和劳森证明，障碍存在的凯勒度量对一个给定的紧凑复杂的流形X的维数n是一个积极的，非零电流的bidegree（n− 1，n− 1）这是（n− 1，n − 1）组成部分的一个d-精确电流的X。一般来说，这样的电流不是d-闭合的，因此闭合正电流的理论不能用来研究它们（尽管一些结果将这个理论扩展到i-闭合的情况，正电流确实存在）。本文的主要结果是，在X是一个在Fujiki类C的流形的情况下，障碍流可以被选择为d-闭的：定理0.1。设X是Fujiki类C中的n维紧致复流形，且不是Kähler流形。则存在一个正的、非零的二次（n− 1，n− 1）的电流T，它是i-正合的。定理0.1直接由定理0.2得出。设X是Fujiki类C中的n维紧致复流形，且X上存在ω a严格正的（1，1）形式使得i ω = 0.则X是一个Kähler流形。这两个定理是Peternell [Pe]所证明的代数情形到解析情形的推广。定理0.2类似于Moishezon的定理，该定理指出Moishezon流形是Kähler流形实际上是投射的。A（1，1）如定理0.2中那样形成ω（即，正定义的，且i是闭的）称为强Kähler挠度规（SKT）。例如，参见[FiTo]以了解SKT指标的介绍。因此，定理0.2指出，在Fujiki类C中支持SKT度量的流形实际上是Kähler。在曲面上，定理0.1和0.2是空的，因为在Fujiki类C中的任何曲面都是Kähler。但是在3重上，定理0.1意味着任何闭合的障碍物都包含一条非零曲线：2012年11月6日由编辑接收。2010年数学学科分类。小学32 J27;中学32 Q15。提交人得到了欧洲共同体第七框架方案内的玛丽·居里国际重返社会补助金和CNCS补助金PN-II-ID-PCE-2011-3-0269。c ©2014 American Mathematical Society 3561许可证或版权限制可能适用于再分发;见https://www.ams.org/journal-terms-of-use
We prove that a manifold in the Fujiki class C which supports a i∂∂̄-closed metric is Kähler. This result implies that on a compact complex manifold in the Fujiki class C which is not Kähler there exists a nonzero i∂∂̄exact, positive current of bidimension (1, 1). Introduction In [HaLa], Harvey and Lawson proved that the obstruction to the existence of a Kähler metric on a given compact complex manifold X of dimension n is a positive, nonzero current of bidegree (n− 1, n− 1) which is the (n− 1, n− 1) component of a d-exact current on X. In general, such currents are not d-closed, therefore the theory of closed positive currents cannot be used to study them (although some results extending this theory to the case of i∂∂̄-closed, positive currents do exist). The main result of this paper is that, in the case when X is a manifold in the Fujiki class C, the obstruction current can be chosen to be d-closed: Theorem 0.1. Let X be a compact complex manifold of dimension n in the Fujiki class C and which is not Kähler. Then there exists a positive, nonzero current T of bidegree (n− 1, n− 1) which is i∂∂̄-exact. Theorem 0.1 follows immediately from Theorem 0.2. Let X be a compact complex manifold of dimension n in the Fujiki class C and suppose there exists ω a strictly positive (1, 1) form on X such that i∂∂̄ω = 0. Then X is a Kähler manifold. The two theorems are generalizations to the analytic case of the algebraic case which was proved by Peternell [Pe]. Theorem 0.2 is similar to Moishezon’s theorem which states that a Moishezon manifold which is Kähler is in fact projective. A (1, 1) form ω as in Theorem 0.2 (i.e., positive defined, and i∂∂̄-closed) is called a strong Kähler with torsion (SKT ) metric. See for instance [FiTo] for an introduction to SKT metrics. Therefore, Theorem 0.2 states that a manifold in Fujiki class C which supports an SKT metric is in fact Kähler. On surfaces, Theorems 0.1 and 0.2 are vacuous since any surface in the Fujiki class C is Kähler. But on 3-folds, Theorem 0.1 implies that any closed obstruction contains a nonzero curve: Received by the editors November 6, 2012. 2010 Mathematics Subject Classification. Primary 32J27; Secondary 32Q15. The author was supported by a Marie Curie International Reintegration Grant within the 7th European Community Framework Programme and the CNCS grant PN-II-ID-PCE-2011-3-0269. c ©2014 American Mathematical Society 3561 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use