Similar to orbit spaces coming from group actions that reduce the symmetry in a system, a symplectic quotient is the result of reducing a Hamiltonian system both with respect to a group action, but also via a given value of a function, called a momentum map. In particular, given an invariant such as energy or angular momentum, one can focus on the pre-image of a value of these functions and apply the reduction of symmetry to the corresponding level set. The result is a new symplectic space.
In the case that the level set is not a submanifold, interesting singularities arise, which become even more interesting after the symmetry reduction. In general, these are not the same types of spaces as orbit spaces of Lie group actions. Indeed, the latter about a singularity looks more like a cone over the quotient of a sphere. In the case of a Hamiltonian circle action, the former about a singularity looks more like a cone over the quotient of a product of spheres. There is the natural question of when these yield the same thing: when can a symplectic quotient be realised as the orbit space of a Lie group action on a manifold. This question is explored in [W:sympl].The above question involves the smooth structure of a symplectic quotient. One can look at other structures, such as a complex structure, or related structures, on the space. In fact, some Hamiltonian group actions on Kähler manifolds are intimately related to group actions of the corresponding complexified group, with the orbit spaces of the latter matching the symplectic quotients of the former in certain circumstances. It turns out this relationship yields a rich theory, especially when singularities are concerned. For example, if one changes the value of the momentum map at which the symplectic quotient is obtained, how does this affect the topology of the resulting quotient, and how does the result compare with the complex orbit space? This has been studied in [GS] and [K] in nicer cases, in which birational equivalence and other qualities are proven. In my paper with Susan Tolman [TW], we consider the case where we do not have a Kähler structure, but only a symplectic structure with a "taming" condition involving a circle action and a complex structure. We show that many of the same qualities persist in this more general world, and Tolman [T] goes on to use these results to answer a (previously open) question of Dusa McDuff: does there exist a non-Hamiltonian circle action on a closed, connected, six-dimensional symplectic manifold with isolated fixed points?
- [GS] Victor Guillemin and Shlomo Sternberg, "Birational equivalence in the symplectic category", Invent. Math. 97 (1989), 485-522.
- [K] Frances Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31. Princeton University Press, 1984.
- [T] Susan Tolman, "Non-Hamiltonian actions with isolated fixed points", Invent. Math. 210 (2017), 877-910.
- [TW] Susan Tolman and Jordan Watts, Tame circle actions, Trans. Amer. Math. Soc., Vol. 369 (2017), 7443-7467.
- [W:sympl] Jordan Watts, Symplectic quotients and representability: the circle action case, 19 pages (preprint).