殆乘积凯勒曲面
Almost-product Kähler surfaces
AI总结：
本文研究凯勒曲面切丛正交分解为两个可积复线性子丛的情形，给出适配该分解的局部坐标、凯勒情形的刻画及相关偏微分方程组的局部可解性证明，还揭示了分解直和项的全纯性与叶的全测地性的等价关系。
AI中文摘要：
我们研究凯勒曲面的切丛正交分解为两个可积复线性子丛的情形。这等价于存在一个单位长度的闭反自对偶2-形式，进而导出一个曲率条件。因此，这类分解未必存在，即使是局部也可能不存在，这与Grant和Vickers[11]证明的一般黎曼四维流形形成对比——在实解析性假设下，一般黎曼四维流形局部总能容纳两个相互正交的可积秩2分布。我们在凯勒曲面中构造了适配上述分解的自然局部坐标，可用于简单构造例子。我们还在由上述坐标导出的更广泛的殆凯勒曲面类中，给出了凯勒情形的刻画，该刻画涉及一个施加于四个四变量未知函数的一阶拟线性偏微分方程组，利用Cartan检验，我们证明了该方程组的局部可解性。最后，我们证明，对于凯勒曲面中的这类分解，其中一个直和项全纯当且仅当另一个具有全测地叶，并描述了具有该性质的例子的一般构造方法。
英文摘要：
We study the case where the tangent bundle of a Kähler surface splits orthogonally into two integrable complex-line subbundles. This amounts to the presence of a unit-length closed anti-self-dual 2-form, leading in turn to a curvature condition. Therefore, such a decomposition need not exist, even locally, in contrast with general Riemannian four-manifolds which, under the assumption of real-analyticity, were shown by Grant and Vickers [11] to always admit, locally, two mutually orthogonal integrable rank-two distributions. We exhibit local coordinates naturally adapted to a decomposition as above in a Kähler surface, which may be used for simple constructions of examples. We also provide a characterization of the Kähler case within the wider class of almost-Kähler surfaces arising from the coordinates just mentioned. The characterization involves a system of four first-order quasilinear partial differential equations imposed on four unknown functions of four variables and, using Cartan's test, we prove the system's local solvability. Finally, we show that, for such a decomposition in a Kähler surface, one of the summands is holomorphic if and only if the other one has totally geodesic leaves, and describe a general construction of examples with this last property.
