Hamiltonian circle action, invariant hypersurface and the complex projective space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917032390295552 |
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| author | Li, Ping |
| author_facet | Li, Ping |
| contents | Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology cell, which is satisfied when $M\setminus D$ is contractible, then $M$ and $D$ are homotopy complex projective spaces with standard Chern classes and the $S^1$-representations on the fixed-point set of $(M,D)$ are the same as those arising from the standard linear actions on $(\mathbb{P}^n,\mathbb{P}^{n-1})$, provided that $n \not \equiv 3 \pmod 4$. This can be viewed as the transformation group analogue to a recent result obtained by Peternell and the author, where the latter was conjectured by Fujita more than four decades ago. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamiltonian circle action, invariant hypersurface and the complex projective space Li, Ping Differential Geometry Algebraic Topology Symplectic Geometry 53D05, 57R20, 32Q60, 37B05, 58J20 Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology cell, which is satisfied when $M\setminus D$ is contractible, then $M$ and $D$ are homotopy complex projective spaces with standard Chern classes and the $S^1$-representations on the fixed-point set of $(M,D)$ are the same as those arising from the standard linear actions on $(\mathbb{P}^n,\mathbb{P}^{n-1})$, provided that $n \not \equiv 3 \pmod 4$. This can be viewed as the transformation group analogue to a recent result obtained by Peternell and the author, where the latter was conjectured by Fujita more than four decades ago. |
| title | Hamiltonian circle action, invariant hypersurface and the complex projective space |
| topic | Differential Geometry Algebraic Topology Symplectic Geometry 53D05, 57R20, 32Q60, 37B05, 58J20 |
| url | https://arxiv.org/abs/2510.19190 |