Mathematics – Symplectic Geometry
Scientific paper
2011-12-12
Mathematics
Symplectic Geometry
52 pages, comments are very welcome
Scientific paper
Tseng and Yau developed primitive cohomology theories on symplectic manifolds. In addition, they proposed a definition of primitive homology, and proved that there is a natural homomorphism from the primitive homology to the primitive cohomology. Inspired by the work of Tseng and Yau, we develop a new approach to the symplectic Hodge theory, and prove in this paper that there is a Poincar\'{e} duality between the primitive homology and cohomology for any compact symplectic manifold with the Hard Lefschetz property. Among other things, we introduce a De Rham complex of real flat chains on symplectic manifolds, and use it to give a dual chain description of the symplectic Hodge adjoint operator. For projective K\"ahler manifolds, the Poincar\'{e} duality between the primitive cohomology and homology provides a new geometric interpretation of primitive cohomology classes from the viewpoint of symplectic Hodge theory, which is very different from the ones what algebraic geometers had before. As an application, we use the primitive version of the Poincar\'e duality theorem to investigate the support of symplectic Harmonic representatives of Thom classes, and provide an answer to a question asked by Guillemin.
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