Twistor theory of symplectic manifolds

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)Sp(2n,R)/U(n) existing over any given symplectic 2n2n-manifold MM. Then, after recalling the construction of the almost complex structure induced on Z by a symplectic connection on MM, we study and find some specific properties of both. We show a few examples of twistor spaces, develop the interplay with the symplectomorphisms of MM, find some results about a natural almost Hermitian structure on Z and finally prove its n+1n+1-holomorphic completeness. We end by proving a vanishing theorem about the Penrose transform.