Existence of twisted constant scalar curvature Kähler metrics with a large twist

Abstract

Suppose that there exist two Kähler metrics $ω$ and $α$ such that the metric contraction of $α$ with respect to $ω$ is constant, i.e. $Λ_ω α= \text{const}$. We prove that for all large enough $R>0$ there exists a twisted constant scalar curvature Kähler metric $ω'$ in the cohomology class $[ ω]$, satisfying $S(ω' ) - R Λ_{ω' } α= \text{const}$. We discuss its implication to $K$-stability and the continuity method recently proposed by X.X. Chen.