Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910987540496384 |
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| author | Mayrand, Maxence |
| author_facet | Mayrand, Maxence |
| contents | The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_09282 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids Mayrand, Maxence Differential Geometry Symplectic Geometry 53D17, 53C26, 53C28, 32G05 The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem. |
| title | Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids |
| topic | Differential Geometry Symplectic Geometry 53D17, 53C26, 53C28, 32G05 |
| url | https://arxiv.org/abs/2011.09282 |