Zeros of random holomorphic sections of big line bundles with continuous metrics
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911837340041216 |
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| author | Bayraktar, Turgay Coman, Dan Marinescu, George Nguyên, Viêt-Anh |
| author_facet | Bayraktar, Turgay Coman, Dan Marinescu, George Nguyên, Viêt-Anh |
| contents | Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeros of random holomorphic sections of big line bundles with continuous metrics Bayraktar, Turgay Coman, Dan Marinescu, George Nguyên, Viêt-Anh Complex Variables Differential Geometry Probability Primary 32L10, Secondary 32A60, 32C20, 32U05, 32U40, 60D05 Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$. |
| title | Zeros of random holomorphic sections of big line bundles with continuous metrics |
| topic | Complex Variables Differential Geometry Probability Primary 32L10, Secondary 32A60, 32C20, 32U05, 32U40, 60D05 |
| url | https://arxiv.org/abs/2404.08116 |