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arXiv · 2412.02564

From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones

Vestislav Apostolov·Abdellah Lahdili·Eveline Legendre
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Abstract

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.

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Vestislav Apostolov, Abdellah Lahdili, Eveline Legendre. 2024-12-03. From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones. https://arxiv.org/abs/2412.02564

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