Twistor theory of symplectic manifolds

Details Twistor theory of symplectic manifolds Twistor theory of symplectic manifolds

2004-05-27

arxiv.org/abs/math/0405516v2

Journal of Geometry and Physics, Volume 56, Issue 2, February 2006, Pages 214-246

Mathematics

Symplectic Geometry

34 pages

Scientific paper

10.1016/j.geomphys.2005.01.007

This article is a contribution to the understanding of the geometry of the twistor space of a symplectic manifold. We consider the bundle $Z$ with fibre the Siegel domain Sp(2n,R)/U(n) existing over any given symplectic 2n-manifold M. Then, after recalling the construction of the almost complex structure induced on $Z$ by a symplectic connection on M, we study and find some specific properties of both. We show a few examples of twistor spaces, develop the interplay with the symplectomorphisms of M, find some results about a natural almost Hermitian structure on $Z$ and finally prove its n+1-holomorphic completeness. We end by proving a vanishing theorem about the Penrose transform.

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