## Journal of Differential Geometry

### Cohomology and Hodge theory on symplectic manifolds: III

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 103, Number 1 (2016), 83-143.

Dates
First available in Project Euclid: 12 April 2016

https://projecteuclid.org/euclid.jdg/1460463564

Digital Object Identifier
doi:10.4310/jdg/1460463564

Mathematical Reviews number (MathSciNet)
MR3488131

Zentralblatt MATH identifier
1353.53085

#### Citation

Tsai, Chung-Jun; Tseng, Li-Sheng; Yau, Shing-Tung. Cohomology and Hodge theory on symplectic manifolds: III. J. Differential Geom. 103 (2016), no. 1, 83--143. doi:10.4310/jdg/1460463564. https://projecteuclid.org/euclid.jdg/1460463564