Symplectic duality of symmetric spaces

• Autores: Antonio J. Di Scala, Andrea Loi
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 217, Nº 5, 2008, págs. 2336-2352
• Idioma: inglés
• DOI: 10.1016/j.aim.2007.10.009
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• Resumen

  • Let be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ?hyp. Denote by (M*,?FS) the compact dual of (M,?hyp), where ?FS is the Fubini�Study form on M*. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ?M, where ?0 is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ?M, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of . As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin