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Some examples of the twistor construction. (English) Zbl 0596.53057
Contributions to several complex variables, Hon. W. Stoll, Proc. Conf. Complex Analysis, Notre Dame/Indiana 1984, Aspects Math. E9, 51-67 (1986).
[For the entire collection see Zbl 0585.00003.]
A hyperkähler manifold X is a Kähler manifold of even dimension with a holomorphic 2-form \(\omega\), everywhere nonsingular, and covariant constant.
Let M be a compact, simply-connected homogeneous Kähler manifold and U its holomorphic isometry group. Then the following theorem is proved: Theorem. There exists a U-invariant hyperkähler metric on a neighborhood of the 0-section in the holomorphic cotangent bundle \(T^*(M)\) of M such that the canonical symplectic form \(\omega_{can}\) on \(T^*(M)\) is covariant constant.
The proof is given by the inversion theorem for the twistor construction in the hyperkähler case.
Reviewer: T.Ochiai

53C55 Global differential geometry of Hermitian and Kählerian manifolds