Fibrations on the 6-sphere and Clemens threefolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910565281038336 |
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| author | Honda, Nobuhiro Viaclovsky, Jeff |
| author_facet | Honda, Nobuhiro Viaclovsky, Jeff |
| contents | Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum $k \# (S^3 \times S^3)$ for $k \geq 2$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fibrations on the 6-sphere and Clemens threefolds Honda, Nobuhiro Viaclovsky, Jeff Algebraic Geometry Complex Variables Differential Geometry Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum $k \# (S^3 \times S^3)$ for $k \geq 2$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$. |
| title | Fibrations on the 6-sphere and Clemens threefolds |
| topic | Algebraic Geometry Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2403.05035 |