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

Abstract

Pl\"ucker coordinates define the $T^n$-equivariant embedding $p : G_{n,2}\to \C P^{N}$ of a complex Grassmann manifold $G_{n,2}$ into the complex projective space $\C P^{N}$, $N=\binom{n}{2}-1$ for the canonical $T^n$-action on $G_{n,2}$ and the $T^n$-action on $\C P^{N}$ given by the second exterior power representation $T^n\to T^{N}$ and the standard $T^{N}$-action. Let $\mu : G_{n,2}\to \Delta_{n,2}\subset \R ^{n}$ and $\tilde{\mu}: \C P^{N}\to \Delta _{n,2}\subset \R^n$ be the moment maps for the $T^n$-actions on $G_{n,2}$ and $\C P^{N}$ respectively, such that $\tilde{\mu} \circ p=\mu$. The preimages $\mu^{-1}({\bf x})$ and $\tilde{\mu} ^{-1}({\bf y})$ are smooth submanifolds in $G_{n, 2}$ and $\C P^{N}$, for any regular values ${\bf x}, {\bf y} \in \Delta _{n,2}$ for these maps, respectively. The orbit spaces $\mu^{-1}({\bf x})/T^n$ and $\tilde{\mu}^{-1}({\bf y})/T^n$ are symplectic manifolds, which are known as symplectic reduction. The regular values for $\mu$ and $\tilde{\mu}$ coincide for $n=4$ and we prove that $\mu^{-1}({\bf x})$ and $\tilde{\mu}^{-1}({\bf x}) $ do not depend on a regular value ${\bf x}\in \Delta_{4,2}$. We provide their explicit topological description, that is we prove $\mu^{-1}({\bf x})\cong S^3\times T^2$ and $\tilde{\mu} ^{-1}({\bf x})\cong S^5\times T^2$. The Deligne - Mumford compactification $\overline{\mathcal{M}}_{0, n}$ is proved to be a symplectic reduction of $G_{n,2}$ by the canonical $T^n$-action if and only if $n=4,5$, while the Losev - Manin compactification is a such symplectic reduction if and only if $n=5$.