A canonical hyperkähler metric on the total space of a cotangent bundle (Q2715739)

In the paper the author gives the simplified proof of the following:NEWLINENEWLINETheorem. Let \((M, g)\) be a Kähler manifold. Then the total space \(X- T^*M\) of the cotangent bundle to \(M\) admits an \(U(1)\)-invariant hyper-Kähler metric \(h\), defined in the formal neighborhood of the zero section \(M\subset T^*M- X\), such that \(h\) restricts to \(g\) on the zero section \(M\subset X\) and the pair \(\{\Omega, h\}\), defines a hyper-Kähler structure on a certain neighborhood of \(M\) in \(X\), where \(\Omega\) is the canonical holomorphic 2-form on \(X\).NEWLINENEWLINE In fact the author presents only the proof of existence of the formal germ of the canonical metric near \(M\subset X\). The complete proof of convergence of this formal germ to the metric on a certain neighborhood of \(M\) is contained in the first version of the proof. This result was first proved by the author in his paper ``hyper-Kähler metrics on total spaces of cotangent bundles'' in \textit{D. Kaledin} and \textit{M. Verbitsky} [Hyper-Kähler manifolds, Math. Phys. Ser. 12, Cambridge MA (1999; Zbl 0990.53048)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].