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 ${\widehat{g}=g/\tau^{2}}$ is a gradient Ricci soliton which has soliton function 1/τ, we show that ${\widehat{g}}$ is Kähler with respect to another complex structure, and locally of a type first described by Koiso, and also Cao. 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 ${(g,\tau) \rightarrow (\widehat{g},1/\tau)}$ .