Hamiltonian 2-forms in Kähler geometry, II Global Classification
Hamiltonian 2-forms in Kähler geometry, II Global Classification
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卡勒几何中的哈密顿量 2 型，II 全球分类
DOI：
10.4310/jdg/1115669513
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发表时间：
2004
影响因子：
2.5
通讯作者：
Christina W. Tonneson
中科院分区：
文献类型：
--
作者：
V. Apostolov;D. Calderbank;P. Gauduchon;Christina W. Tonneson
我们给出了紧Kahler流形的一个分类，
一个汉密尔顿2-形式（这是局部分类的第一部分，
这项工作）。这涉及两个独立利益的组成部分。
第一个是刚性哈密顿环面作用量的概念。这
卡勒流形上环面作用的自然条件是
在第一部分中介绍了当地的情况，但这些行动证明是非常重要的。
在全球范围内表现良好，导致相当明确的分类：
直到一个爆破的，紧凑的Kahler流形，
Hamilton环面作用是环面Kahler流形的丛。
第二个想法是复曲面几何的一个特例，我们称之为
正向的我们证明了正交Kahler流形是单纯的
复射影空间，但我们扩展我们的分析
到正交轨道褶皱，那里的几何形状更丰富。我们
从而得到了Kahler-Einstein 4-orbifolds的新例子。
结合这两个主题，我们证明，紧凑的卡勒
具有Hamilton 2-形式的流形被blow-downs覆盖
的射影丛的Kahler产品，我们明确地描述
具有哈密顿2-形式的Kahler度量是如何
参数化。我们解释了这是如何为构建
极值Kahler度量的新例子-特别是
我们称之为弱Bochner平坦的此类度量的子类。
我们还提供了一个自成一体的治疗理论，
紧复曲面Kahler流形，因为我们需要它，并找到
现有文献不完整。
We present a classification of compact Kahler manifolds admitting
a hamiltonian 2-form (which were classified locally in part I of
this work). This involves two components of independent interest.
The first is the notion of a rigid hamiltonian torus action. This
natural condition, for torus actions on a Kahler manifold, was
introduced locally in part I, but such actions turn out to be remarkably
well behaved globally, leading to a fairly explicit classification:
up to a blow-up, compact Kahler manifolds with a rigid
hamiltonian torus action are bundles of toric Kahler manifolds.
The second idea is a special case of toric geometry, which we call
orthotoric. We prove that orthotoric Kahler manifolds are diffeomorphic
to complex projective space, but we extend our analysis
to orthotoric orbifolds, where the geometry is much richer. We
thus obtain new examples of Kahler–Einstein 4-orbifolds.
Combining these two themes, we prove that compact Kahler
manifolds with hamiltonian 2-forms are covered by blow-downs
of projective bundles over Kahler products, and we describe explicitly
how the Kahler metrics with a hamiltonian 2-form are
parameterized. We explain how this provides a context for constructing
new examples of extremal Kahler metrics—in particular
a subclass of such metrics which we call weakly Bochner-flat.
We also provide a self-contained treatment of the theory of
compact toric Kahler manifolds, since we need it and find the
existing literature incomplete.