Hyperkähler metrics on cotangent bundles (Q2706826)

Let \(X\) be a real analytic Kähler manifold. By using twistor theory, the author proves the existence of a hyper-Kähler metric in a neighborhood of the zero section of the cotangent bundle of \(X\), which is compatible with the canonical holomorphic symplectic structure. Moreover, the \(S^1\)-action given by the scalar multiplication in the fibres is isometric and the restriction of the hyper-Kähler metric to the zero section induces the original Kähler metric on \(X\). The author considers the question of extending the hyper-Kähler metric constructed above to the full cotangent bundle and investigates the completeness of this extension in some special cases.