Cohomology and Hodge Theory on Symplectic Manifolds: III

Chung-Jun Tsai National Taiwan University Li-Sheng Tseng University of California, Irvine Shing-Tung Yau Harvard University

Symplectic Geometry mathscidoc:1702.34001

Distinguished Paper Award in 2017

J. Differential Geometry, 103, 83-143, 2016
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Papers 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.
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  title={Cohomology and Hodge Theory on Symplectic Manifolds: III},
  author={Chung-Jun Tsai, Li-Sheng Tseng, and Shing-Tung Yau},
  booktitle={J. Differential Geometry},
Chung-Jun Tsai, Li-Sheng Tseng, and Shing-Tung Yau. Cohomology and Hodge Theory on Symplectic Manifolds: III. 2016. Vol. 103. In J. Differential Geometry. pp.83-143.
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