Pluripotential geometry on semi-positive effective divisors of numerical dimension one
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911391297830912 |
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| author | Koike, Takayuki |
| author_facet | Koike, Takayuki |
| contents | We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact Kähler manifolds. In one of our main results, for a $\mathbb{Q}$-effective line bundle satisfying a natural torsion-type assumption, we show the equivalence between semi-positivity and semi-ampleness. More generally, for an effective nef divisor of numerical dimension one, we characterize the semi-positivity of the associated line bundle in terms of the existence of a certain type of pseudoflat fundamental system of neighborhoods of the support. Furthermore, for an effective semi-positive divisor, we prove a dichotomy: either the divisor is the pull-back of a $\mathbb{Q}$-divisor by a fibration onto a Riemann surface, or the Hartogs extension phenomenon holds on the complement of its support. Our proof is based on a pluripotential method that has previously been used for studying the boundaries of pseudoconvex domains, which allows us to investigate the complex-analytic structure of neighborhoods of the support of the divisor even when the manifold is non-compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pluripotential geometry on semi-positive effective divisors of numerical dimension one Koike, Takayuki Complex Variables Algebraic Geometry Primary 32J25, Secondary 32U05, 14C20, 32D15 We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact Kähler manifolds. In one of our main results, for a $\mathbb{Q}$-effective line bundle satisfying a natural torsion-type assumption, we show the equivalence between semi-positivity and semi-ampleness. More generally, for an effective nef divisor of numerical dimension one, we characterize the semi-positivity of the associated line bundle in terms of the existence of a certain type of pseudoflat fundamental system of neighborhoods of the support. Furthermore, for an effective semi-positive divisor, we prove a dichotomy: either the divisor is the pull-back of a $\mathbb{Q}$-divisor by a fibration onto a Riemann surface, or the Hartogs extension phenomenon holds on the complement of its support. Our proof is based on a pluripotential method that has previously been used for studying the boundaries of pseudoconvex domains, which allows us to investigate the complex-analytic structure of neighborhoods of the support of the divisor even when the manifold is non-compact. |
| title | Pluripotential geometry on semi-positive effective divisors of numerical dimension one |
| topic | Complex Variables Algebraic Geometry Primary 32J25, Secondary 32U05, 14C20, 32D15 |
| url | https://arxiv.org/abs/2511.13903 |