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May 2016 Cohomology and Hodge theory on symplectic manifolds: III
Chung-Jun Tsai, Li-Sheng Tseng, Shing-Tung Yau
J. Differential Geom. 103(1): 83-143 (May 2016). DOI: 10.4310/jdg/1460463564


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 $A_{\infty}$-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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Chung-Jun Tsai. Li-Sheng Tseng. Shing-Tung Yau. "Cohomology and Hodge theory on symplectic manifolds: III." J. Differential Geom. 103 (1) 83 - 143, May 2016.


Published: May 2016
First available in Project Euclid: 12 April 2016

zbMATH: 1353.53085
MathSciNet: MR3488131
Digital Object Identifier: 10.4310/jdg/1460463564

Rights: Copyright © 2016 Lehigh University


Vol.103 • No. 1 • May 2016
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