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Mathematics Colloquium

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Date:  Tuesday, February 18, 2014

Title:  A Penrose transform for nonembeddable CR Structures on $S^{3}$

Abstract:  A Penrose transform for nonembeddable CR Structures on $S^{3}$ In this talk, we give an explicit correspondence between the space of nonembeddale CR structures on $S^{3}$ and the space of holomorphic projective structures on $\Bbb{B}^{2} \subset \Bbb{C}^{2}$. The space of nonembeddable CR structures near the standard structure on $S^{3}$ can be classified by two conjugate holomorphic functions of two variables. These structures can be naturally extended to deformations of the standard complex structure on $\Bbb{P}^{2}\setminus \Bbb{B}^{2}$ which agree with the standard complex structure on the $\Bbb{P}^{1}$ at $\infty$. Since the resulting pseudoconcave surface has rational curves with self intersection one, Hitchin's nonlinear Penrose transform identifies the space of rational curves as a second complex manifold with a holomorphic projective structure defined by the incidence relation between rational curves. We explicitly exhibit this correspondence between pseudoconcave manifolds and projective structures as a deformation of the standard duality between lines and planes in $\Bbb{P}^{2}$; in doing so, we demonstrate how the conjugate holomorphic functions describing the deformation of the CR structure simultaneously define a deformation of the flat projective structure on $\Bbb{B}^{2}$.


Speaker:  John Bland
Institution:  University of Toronto

 

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