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Reduction of complex Poisson manifolds. (English) Zbl 0898.53024

The notion of reduction of a manifold with some structure first came to the fore in an article of J. Marsden and A. Weinstein [Rep. Math. Phys. 5, 121-130 (1974; Zbl 0327.58005)] in the context of symplectic geometry. As a symplectic manifold has a Poisson structure naturally associated to it, also this Poisson structure was reduced in the progress. This led J. Marsden and T. Ratiu [Lett. Math. Phys. 11, 161-169 (1986; Zbl 0602.58016)] to develop a theory for the reduction of general (real) Poisson manifolds.
In the present article, the author defines the reduction of complex Poisson manifolds and proves a reduction theorem similar to the one in the real case. The ideas, suitably adapted to the complex structure, are similar to the ones in the real context. Moreover, the author shows that a complex Poisson structure is reducible if and only if the underlying real Poisson structure is. The theory is illustrated by the reduction of a complex Poisson structure on \(sl^*(2,\mathbb{C})\).
Reviewer: E.Boeckx (Leuven)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C12 Foliations (differential geometric aspects)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C56 Other complex differential geometry
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