Abstract
AbstractLet M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let ω̄ : 𝒜̂ → M be the relative Albanese over M. 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 M.
Highlights
A compact Kähler manifold admits a holomorphic symplectic form if and only if it admits a hyperKähler structure [1], [10]
We prove that A has a natural holomorphic symplectic structure
Let (X, J, g) be a Ricci– at compact Kähler manifold equipped with a holomorphic symplectic form
Summary
A compact Kähler manifold admits a holomorphic symplectic form if and only if it admits a hyperKähler structure [1], [10]. Let X be a compact manifold equipped with almost complex structures. J , J , J , and let g be a Riemannian metric on X, such that (X, J , J , J , g) is a hyperKähler manifold. T , X −g→ (T , X)* , where TRX ⊗ C = T , X ⊕ T , X is the type decomposition with respect to the almost complex structure J ; J produces a C∞ isomorphism
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