Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914273839546368 |
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| author | Koike, Takayuki |
| author_facet | Koike, Takayuki |
| contents | We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact Kähler manifold. Our approach identifies the obstruction as a global analytic class supported on a neighborhood of $Y$, and relates its vanishing to the solvability of a $\partial\overline{\partial}$-problem on neighborhoods of $Y$. As a consequence we obtain cohomological criteria ensuring the formal principle. We also construct a holomorphic family of compact Kähler surfaces containing a curve with topologically trivial normal bundle in which the formal principle holds for almost every fiber but fails for uncountably many fibers, exhibiting an instability phenomenon in families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold Koike, Takayuki Complex Variables Algebraic Geometry Primary 32J25, Secondary 32F10 We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact Kähler manifold. Our approach identifies the obstruction as a global analytic class supported on a neighborhood of $Y$, and relates its vanishing to the solvability of a $\partial\overline{\partial}$-problem on neighborhoods of $Y$. As a consequence we obtain cohomological criteria ensuring the formal principle. We also construct a holomorphic family of compact Kähler surfaces containing a curve with topologically trivial normal bundle in which the formal principle holds for almost every fiber but fails for uncountably many fibers, exhibiting an instability phenomenon in families. |
| title | Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold |
| topic | Complex Variables Algebraic Geometry Primary 32J25, Secondary 32F10 |
| url | https://arxiv.org/abs/2409.20367 |