关于广义复结构的局部和整体分类
On the local and global classification of generalized complex structures
摘要：
本文研究了广义复几何中的一些局部和全局分类问题。在第一个主题中，我们刻画了广义复流形的局部结构，证明了复点附近的广义复结构是由全纯Poisson结构产生的。在证明中，我们使用一个平滑的牛顿法沿线的纳什，莫泽和康恩。在第二个主题中，我们考虑给定的正则Poisson结构和横向复结构是否来自广义复结构。我们给出了上同调的准则，我们发现了一些反例和一些意想不到的例子，包括一个紧凑的，经常广义复流形附近的辛叶不辛形。在第三个主题中，我们考虑了具有非退化型变的广义复结构，描述了在型变轨迹上诱导的广义Calabi-Yau结构，并证明了在该轨迹附近的一个局部规范形定理。最后，在第四个主题中，我们给出了广义复主丛满足一定横截性条件的分类;在这种情况下，存在广义平坦联络，并且分类涉及到Courant自同构群的单值映射。
英文摘要：
We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton's method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.
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