Irregular Eguchi–Hanson type metrics and their soliton analogues

We verify the extension to the zero section of momentum construction of Kähler–Einstein metrics and Kähler–Ricci solitons on the total space $Y$ of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki–Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to $Y$, restricts to the associated unit circle bundle as a transversely Kähler–Einstein (Sasakian eta‑Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled Sasakian eta‑Einstein metric to the induced metric of the zero section.

Keywords

Eguchi–Hanson metric, Ricci-flat Kähler metric, toric Fano manifold, Sasaki–Einstein manifold

2010 Mathematics Subject Classification

Primary 53C55. Secondary 53C21.

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Received 23 August 2020

Accepted 31 January 2021

Published 11 April 2021