Irregular Eguchi-Hanson type metrics and their soliton analogues

TL;DR

This paper extends the construction of special Kähler metrics and solitons on complex line bundles over toric Fano manifolds, revealing new geometric structures and relationships with Sasaki-Einstein metrics.

Contribution

It demonstrates the extension of momentum construction to irregular cases and describes the geometric interplay between Kähler-Einstein metrics, solitons, and Sasaki-Einstein structures.

Findings

Extension of metrics to the zero section of line bundles.

Existence of Riemannian submersion from Sasakian to base metrics.

Explicit expression of extended metrics compatible with Sasaki-Einstein structures.

Abstract

We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-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 Kaehler-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.