Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929288542945280 |
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| author | Tapiero, Virgile |
| author_facet | Tapiero, Virgile |
| contents | Let $f$ be a holomorphic endomorphism of $\mathbb{P}^2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We prove that if $μ$ has a local product structure with respect to $T$ then (an iterate of) $f$ preserves a local foliation $\mathcal{F}$ on a neighborhood of $\mathrm{Supp}(T )\backslash\mathcal{E}$,where $\mathcal{E}$ denotes the exceptional set of f . If the local foliation $\mathcal{F}$ extends through $\mathcal{E}$,then it extends to $\mathbb{P}^2$ and is an invariant pencil of lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure Tapiero, Virgile Complex Variables Let $f$ be a holomorphic endomorphism of $\mathbb{P}^2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We prove that if $μ$ has a local product structure with respect to $T$ then (an iterate of) $f$ preserves a local foliation $\mathcal{F}$ on a neighborhood of $\mathrm{Supp}(T )\backslash\mathcal{E}$,where $\mathcal{E}$ denotes the exceptional set of f . If the local foliation $\mathcal{F}$ extends through $\mathcal{E}$,then it extends to $\mathbb{P}^2$ and is an invariant pencil of lines. |
| title | Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2403.17023 |