Feix-Kaledin metric on the total spaces of cotangent bundles to K\'ahler quotients
| Autor: | Abasheva, Anna |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Int. Math. Res. Not., 2021, rnab047 |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we study the geometry of the total space $Y$ of a cotangent bundle to a K\"ahler manifold $N$ where $N$ is obtained as a K\"ahler reduction from $\mathbb C^n$. Using the hyperk\"ahler reduction we construct a hyperk\"ahler metric on $Y$ and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion $\tilde Y$ of the space $Y$ is equipped with a structure of a stratified hyperk\"ahler space. We give a necessary condition for the Feix-Kaledin metric to be complete using an observation of R.Bielawski. Pick a complex structure $J$ on $\tilde Y$ induced from quaternions. Suppose that $J\ne\pm I$ where $I$ is the complex structure whose restriction to $Y = T^*N$ is induced by the complex structure on $N$. We prove that the space $\tilde{Y}_J$ admits an algebraic structure and is an affine variety. Comment: 34 pages; fixed two mistakes (see Prop. 2.16 and Subs. 3.1) insignificant to the proof of the main results, improved some statements, added examples |
| Databáze: | arXiv |
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