Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910051440001024 |
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| author | Charton, Isabelle Kessler, Liat Tolman, Susan |
| author_facet | Charton, Isabelle Kessler, Liat Tolman, Susan |
| contents | Let a torus $T$ act on a symplectic manifold $(M,ω)$ with moment map $ϕ$. We say that the Hamiltonian $T$-manifold $(M,ω,ϕ)$ has complexity one if $\frac{1}{2} \dim M - \dim T = 1$, and that it is Kähler if it admits an invariant compatible complex structure. In this paper, we show how the class of Kähler complexity one Hamiltonian $T$-manifolds sits inside the class of complexity one Hamiltonian $T$-manifolds by proving that every compact, connected Kähler complexity one Hamiltonian $T$-manifold has a trivial painting. As a corollary, we show that two tall compact, connected Kähler complexity one Hamiltonian $T$-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, $(M,ω,ϕ)$ is tall exactly if every non-empty fiber $ϕ^{-1}(α)$ contains more than one orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12404 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings Charton, Isabelle Kessler, Liat Tolman, Susan Symplectic Geometry Differential Geometry Primary 53D35, Secondary 53D20, 57S15 Let a torus $T$ act on a symplectic manifold $(M,ω)$ with moment map $ϕ$. We say that the Hamiltonian $T$-manifold $(M,ω,ϕ)$ has complexity one if $\frac{1}{2} \dim M - \dim T = 1$, and that it is Kähler if it admits an invariant compatible complex structure. In this paper, we show how the class of Kähler complexity one Hamiltonian $T$-manifolds sits inside the class of complexity one Hamiltonian $T$-manifolds by proving that every compact, connected Kähler complexity one Hamiltonian $T$-manifold has a trivial painting. As a corollary, we show that two tall compact, connected Kähler complexity one Hamiltonian $T$-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, $(M,ω,ϕ)$ is tall exactly if every non-empty fiber $ϕ^{-1}(α)$ contains more than one orbit. |
| title | Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings |
| topic | Symplectic Geometry Differential Geometry Primary 53D35, Secondary 53D20, 57S15 |
| url | https://arxiv.org/abs/2603.12404 |