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5
votes
3answers
598 views

Hamiltonian circle actions and Lefschetz pencils

Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic ...
17
votes
4answers
1k views

Invariant Polynomials under a Group Action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
2
votes
1answer
212 views

Do subgroups respect the orbit-closure relation?

Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in ...
13
votes
4answers
1k views

How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...