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

Abstract

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

14 pages

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• Compactness of Kähler metrics with bounds on Ricci curvature and I functional
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• Constant scalar curvature Kaehler metrics on ramified Galois coverings
• Twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces