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Geometria Diferencial

Holomorphic symplectic forms and deformations of complex manifolds
Misha Verbitsky

Terça-feira, 8 de janeiro de 2019, 15:30
Sala 236.

Let $\Omega$ be a closed differential form of constant rank on a manifold, and $B$ its annihilator, considered as a sub-bundle in the tangent bundle. Then $B$ is closed under the commutator. This observation can be used to define a complex structure on a manifold. In particular, a closed, complex-valued differential form $\Omega$ on a manifold of real dimension $4n$ defines a complex structure if $\Omega^n\wedge\bar\Omega^n$ is non-degenerate and $\Omega^{n+1}=0$. The form $\Omega$ is holomorphically symplectic with respect to the complex structure defined this way. This observation can be used to obtain explicit deformations of complex structures on holomorphic symplectic manifolds. In particular, one is able to recover the Tate-Shafarevich deformation of a Lagrangian fibration.