Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action

Abstract

We study symplectic reductions arising from the canonical action of the maximal compact torus $T^n$ on the complex Grassmann manifold $G_{n,2}$ as well as those arising from the $T^n$-action on the complex projective space $\C P^{N}$, $N=\binom{n}{2}-1$ for which the Plücker embedding $G_{n,2}\to \C P^{N}$ is $T^n$-equivariant. We investigate the topology of regular level sets of the moment maps and the corresponding symplectic reductions. For $n=4$ we show that the regular level sets of the moment maps do not depend on a regular value. We prove this set to be homeomorphic to $S^3\times T^2$ in the case $G_{4,2}$, while in the case $\C P^5$ it is homeomorphic to $ S^5\times T^2$. We relate our constructions to moduli spaces of weighted pointed stable genus zero curves. The Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$ is proved to arise as a symplectic reduction of $G_{n,2}$ by the canonical $T^n$-action in precisely the cases $n=4,5$. In the case $n\geq 5$, there is well known the Losev-Manin compactification different from Deligne-Mumford and it appears to be this symplectic reduction only in the case $n=5$. In this way we show that for $n=5$, a symplectic reduction depends on a regular value of the moment map.