Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Indranil Biswas; Tomás L. Gómez; André Oliveira

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arXiv · 2010.06483

Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Indranil Biswas·Tomás L. Gómez·André Oliveira

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Abstract

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperK\"ahler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

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BibTeX RIS

Indranil Biswas, Tomás L. Gómez, André Oliveira. 2020-10-13. Complex Lagrangians in a hyperKaehler manifold and the relative Albanese. https://arxiv.org/abs/2010.06483

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