A complex-analytic characterization of Lagrangian immersions in $\mathbb C^n$ with transverse double points
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909913461030912 |
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| author | Gupta, Purvi Sahu, Rudranil |
| author_facet | Gupta, Purvi Sahu, Rudranil |
| contents | Given a compact smooth totally real immersed $n$-submanifold $M\subset\mathbb C^n$ with only finitely many transverse double points, it is known that if $M$ is Lagrangian with respect to some K{ä}hler form on $\mathbb C^n$, then it is rationally convex in $\mathbb C^n$ (Gayet, 2000), but the converse is not true (Mitrea, 2020). We show that $M$ is Lagrangian with respect to some K{ä}hler form on $\mathbb C^n$ if and only if $M$ is rationally convex {\em and} at each double point, the pair of transverse tangent planes to $M$ satisfies the following diagonalizability condition: there is a complex linear transformation on $\mathbb C^n$ that maps the pair to $\left(\mathbb R^n,(D+i)\mathbb R^n\right)$ for some real diagonal $n\times n$ matrix $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A complex-analytic characterization of Lagrangian immersions in $\mathbb C^n$ with transverse double points Gupta, Purvi Sahu, Rudranil Complex Variables Symplectic Geometry 32E20, 53D12 Given a compact smooth totally real immersed $n$-submanifold $M\subset\mathbb C^n$ with only finitely many transverse double points, it is known that if $M$ is Lagrangian with respect to some K{ä}hler form on $\mathbb C^n$, then it is rationally convex in $\mathbb C^n$ (Gayet, 2000), but the converse is not true (Mitrea, 2020). We show that $M$ is Lagrangian with respect to some K{ä}hler form on $\mathbb C^n$ if and only if $M$ is rationally convex {\em and} at each double point, the pair of transverse tangent planes to $M$ satisfies the following diagonalizability condition: there is a complex linear transformation on $\mathbb C^n$ that maps the pair to $\left(\mathbb R^n,(D+i)\mathbb R^n\right)$ for some real diagonal $n\times n$ matrix $D$. |
| title | A complex-analytic characterization of Lagrangian immersions in $\mathbb C^n$ with transverse double points |
| topic | Complex Variables Symplectic Geometry 32E20, 53D12 |
| url | https://arxiv.org/abs/2511.15306 |