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 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.

On cohomological decomposition of generalized-complex structures

“…A counterpart of the Bott-Chern cohomology in the symplectic case was introduced and studied by S.-T. Yau and L.-S. Tseng [36,37,38,35]:…”

“…For this aim, consider again the generalized-complex structure ρ := exp i −e 36 − e 45 ∧ e 1 + i e 2 by Cavalcanti and Gualtieri [15, §5]. For each t ∈ [0, 1], take the B-field B t := exp(π i t) e 35 − e 46 and the β-field β t := − 1 4 exp(π i t) (e 3 − i exp(π i t) e 4 ) (e 5 − i exp(π i t) e 6 ) .…”

“…A symplectic Hodge theory was proposed by J.-L. Brylinski in [9] and further results in this direction were obtained, among others, by O. Mathieu [29], D. Yan [39], V. Guillemin [22], and G. R. Cavalcanti [10]. Recently, L.-S. Tseng and S.-T. Yau introduced and studied some symplectic cohomologies [36,37,38,35] Generalized-complex geometry was introduced by N. Hitchin [23] and studied, among others, by his students M. Gualtieri [20,21] and G. R. Cavalcanti [11] (see also [12]). It provides a unified framework for both symplectic and complex structures.…”

Induced C*-Complexes in Metaplectic Geometry

“…See Tsai at al. [67], Schmid, Vilonen [62], Li [41], Nekovář, Scholl [50], Hain [21] and Albin et al [1] for recent contributions.…”

Cohomologies of Generalized Complex Manifolds and Nilmanifolds

“…In this note, we are interested in the generalized Dolbeault cohomologies (Note that, in the complex case, the generalized ∂ and ∂ operators coincide with the complex operators, and so, up to a change of graduation, the above cohomologies are exactly the Dolbeault and the Bott-Chern cohomologies. In the symplectic case, the generalized Dolbeault cohomology is isomorphic to the de Rham cohomology, and the generalized Bott-Chern cohomology has been studied by L.-S. Tseng and S.-T. Yau, see [17,18,19,16].) More precisely, look at the i-eigenbundle L ⊂ (T M ⊕ T * M ) ⊗ C of J ∈ End((T M ⊕ T * M ) ⊗ C) with the Lie algebroid structure given by the Courant bracket and the projection π : L → T M ⊗ C. Take a generalized holomorphic bundle, that is, a complex vector bundle E with a Lie algebroid connection…”