Numerically flat foliations and holomorphic Poisson geometry
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929589748498432 |
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| author | Druel, Stéphane Pereira, Jorge Vitório Pym, Brent Touzet, Frédéric |
| author_facet | Druel, Stéphane Pereira, Jorge Vitório Pym, Brent Touzet, Frédéric |
| contents | We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact Kähler manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08806 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerically flat foliations and holomorphic Poisson geometry Druel, Stéphane Pereira, Jorge Vitório Pym, Brent Touzet, Frédéric Algebraic Geometry Differential Geometry Symplectic Geometry 32M25, 37F75, 32J27, 53D17 We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact Kähler manifolds. |
| title | Numerically flat foliations and holomorphic Poisson geometry |
| topic | Algebraic Geometry Differential Geometry Symplectic Geometry 32M25, 37F75, 32J27, 53D17 |
| url | https://arxiv.org/abs/2411.08806 |