On the Non-Semipositivity of a Nef and Big Line Bundle on Grauert's Example
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909977703088128 |
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| author | Zhang, Yangyang |
| author_facet | Zhang, Yangyang |
| contents | We study the relation between semipositivity, nefness, and bigness of line bundles on compact Kähler manifolds. Every nef and big line bundle on a compact Kähler manifold $X$ is positive when ${\rm dim}\,X = 1$. Kim constructed an explicit example of a nef and big line bundle that is not semipositive in the case ${\rm dim}\,X \ge 3$. Motivated by a conjecture of Filip and Tosatti, we then focus on the case of dimension two. In this talk, we show that the line bundle on Grauert's example is nef and big but not semipositive, by explicitly computing its first obstruction class, which was originally introduced by Koike as a generalization of the Ueda class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Non-Semipositivity of a Nef and Big Line Bundle on Grauert's Example Zhang, Yangyang Algebraic Geometry Complex Variables 32J25, 14C20 We study the relation between semipositivity, nefness, and bigness of line bundles on compact Kähler manifolds. Every nef and big line bundle on a compact Kähler manifold $X$ is positive when ${\rm dim}\,X = 1$. Kim constructed an explicit example of a nef and big line bundle that is not semipositive in the case ${\rm dim}\,X \ge 3$. Motivated by a conjecture of Filip and Tosatti, we then focus on the case of dimension two. In this talk, we show that the line bundle on Grauert's example is nef and big but not semipositive, by explicitly computing its first obstruction class, which was originally introduced by Koike as a generalization of the Ueda class. |
| title | On the Non-Semipositivity of a Nef and Big Line Bundle on Grauert's Example |
| topic | Algebraic Geometry Complex Variables 32J25, 14C20 |
| url | https://arxiv.org/abs/2512.23446 |