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Working paper

Feix-Kaledin metric on the total spaces of cotangent bundles to Kähler quotients

arxiv.org. math. Cornell University, 2020. No. arXiv:2007.05773.
In this paper we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from Cn. Using the hyperkähler reduction we construct a hyperkähler 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 Y~ of the space Y is equipped with a structure of a stratified hyperkähler 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 Y~ induced from quaternions. Suppose that J≠±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 Y~J admits an algebraic structure and is an affine variety.