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

TL;DR

This paper studies the symplectic reduction of complex Grassmann manifolds and projective spaces under torus actions, explicitly describing the topology of certain preimages and identifying conditions under which classical moduli spaces are symplectic reductions.

Contribution

It provides explicit topological descriptions of the preimages of moment maps and characterizes when certain moduli spaces are realized as symplectic reductions of Grassmannians.

Findings

Preimages are smooth manifolds with explicit topology: $S^3\times T^2$ and $S^5\times T^2$.

Regular values for moment maps coincide for $n=4$, with topological invariants proven.

Deligne-Mumford and Losev-Manin spaces are symplectic reductions for specific $n$ values.

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 $n=5$.