The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918160615079936 |
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| author | Filip, Simion Lesieutre, John Tosatti, Valentino |
| author_facet | Filip, Simion Lesieutre, John Tosatti, Valentino |
| contents | We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01012 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds Filip, Simion Lesieutre, John Tosatti, Valentino Algebraic Geometry Dynamical Systems 14C20, 14E07, 14J32, 20F55, 37F80, 57M50 We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold. |
| title | The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds |
| topic | Algebraic Geometry Dynamical Systems 14C20, 14E07, 14J32, 20F55, 37F80, 57M50 |
| url | https://arxiv.org/abs/2312.01012 |