Paper · May 5, 2023

Deformations of Fano manifolds with weighted solitons

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Abstract

We consider weighted solitons on Fano manifolds which include Kaehler-Ricci solitons, Mabuchi solitons and base metrics which induce Calabi-Yau cone metrics outside the zero sections of the canonical line bundles (Sasaki-Einstein metrics on the associated $U (1)$ -bundles). In this paper, we give a condition for a weighted soliton on a Fano manifold $M_{0}$ to extend to weighted solitons on small deformations $M_{t}$ of the Fano manifold $M_{0}$ . More precisely, we show that all the members $M_{t}$ of the Kuranishi family of a Fano manifold $M_{0}$ with a weighted soliton have weighted solitons if and only if the dimensions of $T$ -equivariant automorphism groups of $M_{t}$ are equal to that of $M_{0}$ , and also if and only if the $T$ -equivariant automorphism groups of $M_{t}$ are all isomorphic to that of $M_{0}$ , where the weight functions are defined on the moment polytope of the Hamiltonian $T$ -action. This generalizes a result of Cao-Sun-Yau-Zhang for Kaehler-Einstein metrics.

Comments: Modified along the referee reports. References added. To appear in Pure and Applied Mathematics Quarterly

Ricci solitons Kähler–Ricci solitons Einstein metrics Ricci-flat & Calabi–Yau Kähler geometry Kähler–Einstein & K-stability Sasakian & contact geometry

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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11B.1
Ricci SolitonsBook 11B · Solitons, Compactness and Singularities
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
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The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
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Deformations of Fano manifolds with weighted solitons

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