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Birational equivalence in the symplectic category. (English) Zbl 0683.53033
The geometry of the Marsden-Weinstein reduced phase spaces is studied for symplectic actions of the torus \(S^ 1\times S^ 1\). The first part of the article studies the special but typical case of “blowing up” \({\mathbb{C}}^ n\) at 0 as a symplectic manifold with reduction by \(S^ 1\times S^ 1\). It is shown that even if the level set has singularities the reduced phase space is nonsingular. Using tubular neighborhood techniques these constructions are globalized to general symplectic manifolds. Two related constructions - “ramified coverings” and “real blowing up” - are used to show that critical levels can be split into simple critical levels. The methods of the first part provide a normal form for simple critical points. This gives a description of the change of the symplectic structure of the reduced phase space \(M_ a\) as a passes through critical levels.
The article motivates the notion of symplectic cobordism and the main results can be formulated: Every symplectic cobordism factors into a pair of simple cobordisms and there is a canonical symplectic model for simple cobordisms. The results can be made invariant under any compact group commuting with the \(S^ 1\times S^ 1\) action.
Reviewer: Ch.G√ľnther

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57S25 Groups acting on specific manifolds
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