Calabi’s Conjecture of the Kähler–Ricci Soliton Type

In this paper, we discuss Calabi’s equation of the Kähler–Ricci soliton type on a compact Kähler manifold. This equation was introduced by Zhu as a generalization of Calabi’s conjecture. We give necessary and sufficient conditions for the unique existence of a solution for this equation on a compact Kähler manifold with a holomorphic vector field which has a zero point. We also consider the case of a nowhere vanishing holomorphic vector field, and give sufficient conditions for the unique existence of a solution for this equation.