# Complex Lagrangians in a hyperKähler manifold and the relative Albanese

### Complex Manifolds

#### Article

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>M</jats:italic> be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold <jats:italic>X</jats:italic>, and let ω̄ : 𝒜̂ → <jats:italic>M</jats:italic> be the relative Albanese over <jats:italic>M</jats:italic>. We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over <jats:italic>M</jats:italic>.</jats:p>

Indranil Biswas

Tomás Gómez

### Publication

Year of publication: 2020

Volume: 7

Issue: 1

Pages: 230--240

ISSN: 2300-7443