Moscow Mathematical Journal

Volume 26, Issue 3, July–September 2026 pp. 303–332.

Smooth Manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ Defined by Symplectic Reductions of $T^n$-Action

Authors: Victor M. Buchstaber (1) and Svjetlana Terzić (2)
Author institution:(1) Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street 8, 119991 Moscow, Russia
(2) Faculty of Science and Mathematics, University of Montenegro, Dzordza Vasingtona bb, 81000 Podgorica, Montenegro

Summary:

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 $\mathbb{C} P^{N}$, $N=\binom{n}{2}-1$, for which the Plücker embedding $G_{n,2}\to \mathbb{C} P^{N}$ 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 $\mathbb{C} P^5$ 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.

2020 Math. Subj. Class. Primary: 14L30, 53D20, 57R19; Secondary: 14M25, 57R15.

Keywords: Complex Grassmann manifolds, torus action, symplectic reduction, Deligne–Mumford, Losev–Manin compactifications.

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