Special Kähler-Ricci potentials and Ricci solitons

Abstract

On a manifold of dimension at least six, let $(g,τ)$ be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function $τ$. Off the zero set of $τ$, if the metric $\hat{g}=g/τ^2$ is a gradient Ricci soliton which has soliton function $1/τ$, we show that $\hat{g}$ is Kähler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, $τ$ is a special Kähler-Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci-Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs $(g,τ)$ satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map $(g,τ)\to (\hat{g},1/τ)$.