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Seminários do IMPA

Álgebra

Título |
On the characteristic foliation on smooth hypersurfaces in a holomorphic symplectic manifold |

Expositor |
Ekaterina Amerik
Université de Paris-Sud, França |

Data |
Quarta-feira, 17 de janeiro de 2018, 15:30 |

Local |
Sala 228 |

Resumo |

Let $X$ be an irreducible holomorphic symplectic manifold, that is, a simply-connected manifold such that the space of holomorphic two-forms on $X$ is generated by a symplectic form $\sigma$. If $Y$ is a smooth hypersurface on $X$, the kernel of the restriction of $\sigma$ defines a smooth foliation in curves on $Y$. One can ask when is a general leaf of this foliation Zariski-dense. If $Y$ is uniruled it is easy to see that the leaves are rational curves. The second obvious case when the leaves are degenerate is when $X$ admits a holomorphic lagrangian fibration and $Y$ is the inverse image of a hypersurface on its base. I shall explain two results in this direction: the first one, joint with F. Campana, affirms that a general leaf is not an algebraic curve unless $Y$ is uniruled, and the second one, joint with L. Guseva, concerns the case of dimension 4 and states that the general leaves are Zariski dense unless $Y$ is uniruled or is the inverse image of a hypersurface on the base of a Lagrangian fibration.