Currents, primitive cohomology classes and symplectic Hodge theory

Mathematics – Symplectic Geometry

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Currents, primitive cohomology classes and symplectic Hodge theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Currents, primitive cohomology classes and symplectic Hodge theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Currents, primitive cohomology classes and symplectic Hodge theory will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-708360

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.