The volume of pseudoeffective line bundles and partial equilibrium
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866914231231709184 |
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| author | Darvas, Tamás Xia, Mingchen |
| author_facet | Darvas, Tamás Xia, Mingchen |
| contents | Let $(L,he^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact Kähler manifold $X$. Let $h^0(X,L^k\otimes \mathcal I(ku))$ be the dimension of the space of sections $s$ of $L^k$ such that $h^k(s,s)e^{-ku}$ is integrable. We show that the limit of $k^{-n}h^0(X,L^k\otimes \mathcal I(ku))$ exists, and equals the non-pluripolar volume of $P[u]_\mathcal I$, the $\mathcal I$-model potential associated to $u$. We give applications of this result to Kähler quantization: fixing a Bernstein-Markov measure $ν$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Ampère measure of $P[u]_\mathcal I$, the partial equilibrium. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03827 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The volume of pseudoeffective line bundles and partial equilibrium Darvas, Tamás Xia, Mingchen Differential Geometry Algebraic Geometry Complex Variables Let $(L,he^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact Kähler manifold $X$. Let $h^0(X,L^k\otimes \mathcal I(ku))$ be the dimension of the space of sections $s$ of $L^k$ such that $h^k(s,s)e^{-ku}$ is integrable. We show that the limit of $k^{-n}h^0(X,L^k\otimes \mathcal I(ku))$ exists, and equals the non-pluripolar volume of $P[u]_\mathcal I$, the $\mathcal I$-model potential associated to $u$. We give applications of this result to Kähler quantization: fixing a Bernstein-Markov measure $ν$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Ampère measure of $P[u]_\mathcal I$, the partial equilibrium. |
| title | The volume of pseudoeffective line bundles and partial equilibrium |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2112.03827 |