On Segre-degenerate Levi-flat hypervarieties
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908649050341376 |
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| author | Lebl, Jiří Mernik, Luka |
| author_facet | Lebl, Jiří Mernik, Luka |
| contents | We prove that a singular real-analytic Levi-flat hypersurface $H$ in $\mathbb C^n$ being Segre-degenerate at a point $p$ is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects $H$ at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of $H$ at $p$. The existence of such two-sided support provides families of analytic discs attached to $H$ that covers a neighborhood of $p$. The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to a fixed neighborhood of $p$. Second, the rational hull of $H$ is a neighborhood of $p$, and thus no Levi-flat Segre-degenerate hypersurface in $\mathbb C^n$ can be rationally convex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07259 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Segre-degenerate Levi-flat hypervarieties Lebl, Jiří Mernik, Luka Complex Variables 32C07 (Primary), 32E20 32D10 14P15 (Secondary) We prove that a singular real-analytic Levi-flat hypersurface $H$ in $\mathbb C^n$ being Segre-degenerate at a point $p$ is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects $H$ at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of $H$ at $p$. The existence of such two-sided support provides families of analytic discs attached to $H$ that covers a neighborhood of $p$. The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to a fixed neighborhood of $p$. Second, the rational hull of $H$ is a neighborhood of $p$, and thus no Levi-flat Segre-degenerate hypersurface in $\mathbb C^n$ can be rationally convex. |
| title | On Segre-degenerate Levi-flat hypervarieties |
| topic | Complex Variables 32C07 (Primary), 32E20 32D10 14P15 (Secondary) |
| url | https://arxiv.org/abs/2306.07259 |