Nonregular or nonfree symplectic quotients need not be manifolds

The reduction theorem of this page assumes that the value of the moment map is regular and that the stabilizer acts freely and properly on the level (Marsden--Weinstein--Meyer symplectic reduction). Both hypotheses are load-bearing, and nothing on this page asserts a smooth quotient without them:

• if the value is not regular, the image of the differential is only $\mathrm{ann}({\mathfrak{g}}_p)$ and the level need not be a submanifold of the ambient symplectic manifold at all (The differential of the moment map and the orbit-orthogonal identity);
• if the action on the level is not free, the quotient is only an orbifold or a stratified space in general. For positive coprime integers $k,\mathrm{\ell }$, the effective weighted circle action $e^{i\theta }\cdot (z_1,z_2)=(e^{ik\theta }{z}_1,e^{i\mathrm{\ell }\theta }{z}_2)$ has a level whose quotient of a level is a weighted projective space, and the stabilizers of the coordinate axes produce cone points of orders $k$ and $\mathrm{\ell }$; the classical teardrop and football orbifolds arise this way (da Silva, §24.5).

The counterexample cex-zero-angular-momentum-level-with-nonfree-points-is-singular on the companion examples page exhibits the failure of freeness on the zero angular-momentum level. Singular reduction, slice normal forms, orbifold structures and the stratified symplectic category are deferred to later development; they are named here as boundaries of the present theorem and are not used as suppliers anywhere on this page.