Deformations of Fano manifolds with weighted solitons

We consider weighted solitons on Fano manifolds which include Kähler–Ricci solitons, Mabuchi solitons and base metrics inducing 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 Kähler–Einstein metrics.