On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of 𝑇*{ℂ}ⁿ

Cortés, Vicente

doi:10.1090/S0002-9947-98-02156-4

On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of $T^\ast \mathbb \{C\}^n$
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Trans. Amer. Math. Soc. 350 (1998), 3193-3205 Request permission

Abstract:

For any Lagrangian Kähler submanifold $M \subset T^*\mathbb {C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature $(2,2n)$.

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