Ambitoric几何I：爱因斯坦度量和极值ambikaehler结构
Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
摘要：
我们提出了一个局部分类的共形等价，但相反取向的4维Kaehler度量复曲面相对于一个共同的2-环面行动。在一般的情况下，这些“ambitoric”结构有一个有趣的局部几何依赖于一个二次多项式q和任意函数A和B的一个变量。我们使用这种描述来分类爱因斯坦4度量这是埃尔米特相对于两个方向，以及一类解决方案的爱因斯坦-麦克斯韦方程组，包括黎曼类似物的Plebanski-Demianski度量。我们的分类可以被看作是一个黎曼的结果在相对论由于R。Debever，N. Kamran和R. McLenaghan，是自对偶Einstein厄米四维流形分类的自然扩张，由R.布莱恩特和第一、第三作者。这些Einstein度规正是Bach张量为零的双曲结构，因此具有与之相关的复曲面Kaehler度规为极值的性质（在E.卡拉比）。我们的主要结果也分类后者，提供了新的例子显式极值Kaehler度量。对于Einstein-Maxwell和极值双曲结构，A和B都是四次多项式，但对系数有不同的条件。在本文的续集中，我们考虑全球的例子，并使用它们来解决的存在性问题的极值Kaehler度量复曲面4-orbifolds第二贝蒂数b2=2。
英文摘要：
We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.
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