Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914048109445120 |
|---|---|
| author | Buchstaber, Victor M. Terzić, Svjetlana |
| author_facet | Buchstaber, Victor M. Terzić, Svjetlana |
| contents | Plücker 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 $μ: G_{n,2}\to Δ_{n,2}\subset \R ^{n}$ and $\tildeμ: \C P^{N}\to Δ_{n,2}\subset \R^n$ be the moment maps for the $T^n$-actions on $G_{n,2}$ and $\C P^{N}$ respectively, such that $\tildeμ \circ p=μ$. The preimages $μ^{-1}({\bf x})$ and $\tildeμ ^{-1}({\bf y})$ are smooth submanifolds in $G_{n, 2}$ and $\C P^{N}$, for any regular values ${\bf x}, {\bf y} \in Δ_{n,2}$ for these maps, respectively. The orbit spaces $μ^{-1}({\bf x})/T^n$ and $\tildeμ^{-1}({\bf y})/T^n$ are symplectic manifolds, which are known as symplectic reduction.
The regular values for $μ$ and $\tildeμ$ coincide for $n=4$ and we prove that $μ^{-1}({\bf x})$ and $\tildeμ^{-1}({\bf x}) $ do not depend on a regular value ${\bf x}\in Δ_{4,2}$. We provide their explicit topological description, that is we prove $μ^{-1}({\bf x})\cong S^3\times T^2$ and $\tildeμ ^{-1}({\bf x})\cong S^5\times T^2$.
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 $n=5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action Buchstaber, Victor M. Terzić, Svjetlana Algebraic Topology 14L30, 53D20, 57R19 (primary), 14M25, 57R15 (secondary) Plücker 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 $μ: G_{n,2}\to Δ_{n,2}\subset \R ^{n}$ and $\tildeμ: \C P^{N}\to Δ_{n,2}\subset \R^n$ be the moment maps for the $T^n$-actions on $G_{n,2}$ and $\C P^{N}$ respectively, such that $\tildeμ \circ p=μ$. The preimages $μ^{-1}({\bf x})$ and $\tildeμ ^{-1}({\bf y})$ are smooth submanifolds in $G_{n, 2}$ and $\C P^{N}$, for any regular values ${\bf x}, {\bf y} \in Δ_{n,2}$ for these maps, respectively. The orbit spaces $μ^{-1}({\bf x})/T^n$ and $\tildeμ^{-1}({\bf y})/T^n$ are symplectic manifolds, which are known as symplectic reduction. The regular values for $μ$ and $\tildeμ$ coincide for $n=4$ and we prove that $μ^{-1}({\bf x})$ and $\tildeμ^{-1}({\bf x}) $ do not depend on a regular value ${\bf x}\in Δ_{4,2}$. We provide their explicit topological description, that is we prove $μ^{-1}({\bf x})\cong S^3\times T^2$ and $\tildeμ ^{-1}({\bf x})\cong S^5\times T^2$. 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 $n=5$. |
| title | Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action |
| topic | Algebraic Topology 14L30, 53D20, 57R19 (primary), 14M25, 57R15 (secondary) |
| url | https://arxiv.org/abs/2507.04582 |