Cohomology and Hodge theory on symplectic manifolds: III

Abstract

arXiv:1402.0427v2 [math.SG] 2 May 2014 Cohomology and Hodge Theory on Symplectic Manifolds: III Chung-Jun Tsai, Li-Sheng Tseng and Shing-Tung Yau February 3, 2014 Abstract 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 ∞ -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. Contents 1 Introduction 2 Preliminaries Operations on differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Filtered forms and differential operators . . . . . . . . . . . . . . . . . . . . . . . 14 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16