TY - JOUR
T1 - Symplectic realizations of holomorphic Poisson manifolds
AU - Broka, Damien
AU - Xu, Ping
N1 - Publisher Copyright:
© 2022 International Press of Boston, Inc.. All rights reserved.
PY - 2022
Y1 - 2022
N2 - Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold (X, π) with underlying real smooth manifold X, we prove that there exists a holomorphic symplectic structure in a neighborhood Y of the zero section of T ∗X such that the projection map is a holomorphic symplectic realization of the given holomorphic Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on Y ⊆ T ∗X.
AB - Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold (X, π) with underlying real smooth manifold X, we prove that there exists a holomorphic symplectic structure in a neighborhood Y of the zero section of T ∗X such that the projection map is a holomorphic symplectic realization of the given holomorphic Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on Y ⊆ T ∗X.
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U2 - 10.4310/MRL.2022.V29.N4.A1
DO - 10.4310/MRL.2022.V29.N4.A1
M3 - Article
AN - SCOPUS:85152101969
SN - 1073-2780
VL - 29
SP - 903
EP - 944
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 4
ER -