Twistor spaces of Hyperkähler manifolds with \(S^{1}\)-actions.
A hyper-Kähler structure consists of a triple of anti-commuting complex structures compatible with a fixed metric which is Kähler for all of them. Such manifolds admit a natural twistor space construction. In the present paper the author considers the structure of hyper-Kähler manifolds \(M\) admitting an \(S^1\)-action with the following properties: It is holomorphic with respect to one of the stuctures, scales the Kähler forms of the other two and its fixed point set is of dimension \({1\over 2}(\dim M)\). Then, there is a neighborhood of the fixed point set \(X\), such that the hyper-Kähler structure looks locally as a one on the cotangent bundle \(T^*X\) which is compatible with its holomorphic symplectic structure. Moreover, such structure is unique locally. The results are proven using analysis of the fixed point set and the structure of the corresponding twistor space.
scientific article; zbMATH DE number 1970810
scientific article