Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action (2507.04582v1)

Abstract: Pl\"ucker coordinates define the $T^n$-equivariant embedding $p : G_{n,2}\to \mathbb{C} P^{N}$ of a complex Grassmann manifold $G_{n,2}$ into the complex projective space $\mathbb{C} P^{N}$, $N=\binom{n}{2}-1$ for the canonical $T^n$-action on $G_{n,2}$ and the $T^n$-action on $\mathbb{C} P^{N}$ given by the second symmetric power representation $T^n\to T^{N}$ and the standard $T^{N}$-action. Let $\mu : G_{n,2}\to \Delta_{n,2}\subset \mathbb{R} ^{n}$ and $\tilde{\mu}: \mathbb{C} P^{N}\to \Delta {n,2}\subset \mathbb{R}^n$ be the moment maps for the $T^n$-actions on $G{n,2}$ and $\mathbb{C} P^{N}$ 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 $\mathbb{C} P^{N}$, 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 $\tilde{\mu} ^{-1}({\bf x})\cong S^5\times T^2$. In addition, we discuss, from the point of view of symplectic reduction, our results on description of the orbit space $G_{n,2}/T^n$, which are related to the Deligne-Mumford and Losev-Manin compactifications.