Kodaira dimensions of almost complex manifolds, I
Kodaira dimensions of almost complex manifolds, I
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DOI：
10.1353/ajm.2023.0011
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发表时间：
2018-08
影响因子：
1.7
通讯作者：
Haojie Chen;Weiyi Zhang
中科院分区：
文献类型：
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作者：
Haojie Chen;Weiyi Zhang
这是一系列论文中的第一篇，在这些论文中，我们研究了紧致几乎复杂流形上的多生，Kodaira维数和更一般的Iitaka维数。在几乎复杂流形的Hodge理论的基础上，引入紧致几乎复杂流形上的plurigenera、Kodaira维数和Iitaka维数。我们证明了在几乎复杂范畴中，至少在维$4$中，多生和Kodaira维以及不规则是双分不变量，其中双分态映射被定义为一次伪全纯映射。然而，它们不再是变形不变量，即使在维度$4$或在驯服假设下也是如此。在建立双分不变性的方法上，我们用圆盘叶化技术证明了在几乎复杂的情况下Hartogs的扩展定理。通过实例说明了这些不变量的一些有趣现象。特别地，我们构造了具有大Kodaira维数的不可积紧致几乎复流形。对六球S^6$上的标准几乎复杂结构计算了不同于假设复杂结构的霍奇数和多生数。
This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that the plurigenera and the Kodaira dimension as well as the irregularity are birational invariants in almost complex category, at least in dimension $4$, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension $4$ or under tameness assumption. On the way to establish the birational invariance, we prove the Hartogs extension theorem in the almost complex setting by the foliation-by-disks technique. Some interesting phenomena of these invariants are shown through examples. In particular, we construct non-integrable compact almost complex manifolds with large Kodaira dimensions. Hodge numbers and plurigenera are computed for the standard almost complex structure on the six sphere $S^6$, which are different from the data of a hypothetical complex structure.