From Kähler Ricci solitons to Calabi-Yau Kähler cones

We show that if \(X\) is a smooth Fano manifold which caries a K\"ahler Ricci soliton, then the canonical cone of the product of \(X\) with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and...

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Published in:arXiv.org 2024-12
Main Authors: Apostolov, Vestislav, Lahdili, Abdellah, Legendre, Eveline
Format: Article
Language:English
Subjects:
Solitary waves
Weighting functions
Online Access:Get full text
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Summary:We show that if \(X\) is a smooth Fano manifold which caries a K\"ahler Ricci soliton, then the canonical cone of the product of \(X\) with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions \(v\) over the momentum polytope of a given smooth Fano manifold, for which a \(v\)-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a K\"ahler Ricci soliton and a Fujita type volume bound for the existence of a \(v\)-soliton.
ISSN:2331-8422