基于CR几何的四维流形的交形式和伴随公式
Intersection Forms and the Adjunction Formula for Four-manifolds via CR Geometry
摘要：
这主要是一个简要说明表明，早期的工作赖CR几何提供了一个干净的解释，在高斯地图方面，一个附加公式嵌入表面在一个几乎复杂的四个流形。我们将看到，如果F是几乎复四维流形M中亏格为g的曲面，则2 - 2g + \cdot F - i^{*} c_{1}（M）-\cdot C = 0，其中C是M上的一个双圈，从格拉斯曼Gr（2，其中i^{*} c_{1}（M）是M的第一陈类到F的限制.感兴趣的关键新项是F \cdot C，它将捕获F的点，这些点的切面从周围流形M的殆复结构继承复结构。这些复杂的跳跃点，然后确定在H_{2}（M，Z）的同调类的光滑代表的亏格。进一步地，通过极化，我们可以从已知M的曲面的复跳跃点的性质出发，利用这个公式来确定M上的相交形式。
英文摘要：
This is primarily an expository note showing that earlier work of Lai on CR geometry provides a clean interpretation, in terms of a Gauss map, for an adjunction formula for embedded surfaces in an almost complex four manifold. We will see that if F is a surface with genus g in an almost complex four-manifold M, then 2 - 2 g + F \cdot F - i^{*} c_{1}(M) - 2 F\cdot C = 0, where C is a two-cycle on M pulled back from the cycle of two planes with complex structure in a Grassmannian Gr (2, C^N) via a Gauss map and where i^{*} c_{1}(M) is the restriction of the first Chern class of M to F. The key new term of interest is F \cdot C, which will capture the points of F whose tangent planes inherit a complex structure from the almost complex structure of the ambient manifold M. These complex jump points then determine the genus of smooth representative of a homology class in H_{2}(M, Z). Further, via polarization, we can use this formula to determine the intersection form on M from knowing the nature of the complex jump points of M's surfaces.
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