任意几何II：极值复曲面和Einstein 4-orbifold
Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
摘要：
我们给出了复曲面4-orbifolds M上第二贝蒂数b2（M）=2的极值Kaehler度量的存在性问题的一个显式解决方案.更精确地说，我们证明了M容许这样一个度量当且仅当它的有理Delzant多面体（这是一个标号四边形）在相对复曲面意义下是K-多稳定的（如S. Donaldson，E.勒让德湾Szekelyhidi等人）。此外，在这种情况下，极值Kaehler度量是ambitoric的，即，兼容的共形等价，相反取向的复曲面Kaehler度量，这原来是极值以及。这些结果为标号四边形的K-稳定性提供了一个计算检验。极值的ambitoric结构进行了局部分类的第一部分，这项工作，但在这里，我们只使用直接的事实，即显式Kaehler度量获得有极值，并确定巴赫平坦（共形爱因斯坦）的例子。使用我们的整体结果，后者产生可数无限家庭的紧凑复曲面巴赫平坦Kaehler orbifolds，包括例子，这是全球共形爱因斯坦，和例子是共形的完整光滑爱因斯坦度量的一个开子集，从而扩展了许多作者的工作。
英文摘要：
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.
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