We describe the geometry and the topology of generalized polygon spaces with “edges” in projective spaces, and fixed "edge lengths".
These spaces are symplectic quotients of degenerate co-adjoint orbits on Lie algebras, satisfying a "closing condition". They can also be viewed as moduli spaces of quiver representations of a star-shaped quiver, and as spaces of parabolic bundles over the Riemann sphere. Concentrating on the su(n) case, and using wall-crossing methods (also called flips), we obtain a recursive formula for their Poincaré polynomials.