Existence of twisted constant scalar curvature K\"ahler metrics with a large twist

TL;DR

The paper proves the existence of twisted constant scalar curvature Kähler metrics with large twist parameters, advancing understanding of their existence conditions and implications for K-stability and the continuity method.

Contribution

It establishes the existence of twisted cscK metrics with large twist R, connecting geometric analysis with stability conditions and the continuity method.

Findings

Existence of twisted cscK metrics for large R

Implications for K-stability

Relevance to Chen's continuity method

Abstract

Suppose that there exist two K\"ahler 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 K\"ahler metric $\omega'$ in the cohomology class $[ \omega ]$, satisfying $S(\omega' ) - R \Lambda_{\omega' } \alpha = \text{const}$. We discuss its implication to $K$-stability and the continuity method recently proposed by X.X. Chen.