Cohomology and Hodge Theory on Symplectic Manifolds: III
Cohomology and Hodge Theory on Symplectic Manifolds: III
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DOI：
10.4310/jdg/1349292670
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发表时间：
2009-09
期刊：
影响因子：
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通讯作者：
L. Tseng;S. Yau
中科院分区：
文献类型：
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作者：
L. Tseng;S. Yau
我们引进了辛流形上微分形式的滤子上同调。他们推广和包括讨论的上同调论文I和II作为一个子集。过滤的上同调是有限维的，可以与微分椭圆复形。在代数上，我们证明了过滤上同调给出了Lefschetz映射的双边分辨率，从而，它们与Lefschetz映射的核和上核直接相关。我们还介绍了一种新的，非结合的产品操作的微分形式辛流形。这个乘积在过滤上同调的基础形式上生成一个A-无穷代数结构，并给它们一个环结构。作为一个应用程序，我们演示了如何过滤上同调的环结构可以区分不同的辛四流形的情况下，一个圆倍纤维三流形。
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.