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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.18491 |
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| _version_ | 1866911659312807936 |
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| author | Li, Mu-Lin Liu, Xiao-Lei |
| author_facet | Li, Mu-Lin Liu, Xiao-Lei |
| contents | Let $π\cln X\to Δ^m$ be a proper smooth Kähler morphism from a complex manifold $X$ to the unit polydisc $Δ^m$. Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective manifold $S$. If the canonical line bundle of $S$ is semiample, then we show that all fibers over $Δ^m$ are biholomorphic to $S$. As an application, we obtain that for smooth families where the canonical line bundle of the generic fiber is semiample, birational isotriviality is equivalent to isotriviality. Moreover, we establish a new Parshin-Arakelov type isotriviality criterion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformation rigidity for projective manifolds and isotriviality of smooth families Li, Mu-Lin Liu, Xiao-Lei Algebraic Geometry Let $π\cln X\to Δ^m$ be a proper smooth Kähler morphism from a complex manifold $X$ to the unit polydisc $Δ^m$. Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective manifold $S$. If the canonical line bundle of $S$ is semiample, then we show that all fibers over $Δ^m$ are biholomorphic to $S$. As an application, we obtain that for smooth families where the canonical line bundle of the generic fiber is semiample, birational isotriviality is equivalent to isotriviality. Moreover, we establish a new Parshin-Arakelov type isotriviality criterion. |
| title | Deformation rigidity for projective manifolds and isotriviality of smooth families |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2407.18491 |