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On the base of a lagrangian fibration for a compact hyperkahler manifold

Fedor Bogomolov
Courant and HSE
Thursday, June 13, 2019 - 14:00

In my talk I will discuss our proof with N. Kurnosov that the base of such fibration for complex projective hyperkahler manifold of complax  dimension $4$ is always a projective plane $P^2$. In fact we show that the base of such fibration can not have a singular point of type $E_8$. It was by the theorem of Matsushita and others that only quotient singularities can occur and if the base is smooth then the it is isomorphic to $P^2$. The absence of other singualrities apart from $E_8$ has been already known and we show that $E-8$ can not occur either. Our method can be applied to other types of singularitiesfor the study of  lagrangian fibrations in higher dimensions. More recently similar result was obtained by Huybrechts and Xu.

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