Theorem 11.26.3AFF6DB5A7BE1

Proof. By theorem 11.4.2, choose a conformal Gauduchon form $\gamma =\mathrm e^ a\omega $. Then $K_{\gamma ,h}=\mathrm e^{-a}K_{\omega ,h}\geq 0$. Choose a closed coordinate ball $Q\subset U$ and a smooth function $\chi \geq 0$ supported in its interior, with $\int _ X\chi \gamma ^ n>0$, and set

Thus $\rho =1$ on $X\setminus Q$ and $\int _ X\rho \gamma ^ n=0$. Since $\gamma $ is Gauduchon, proposition 11.23.2, applied with $\gamma $ in place of $\omega $ and $u=0$, gives a smooth real solution $f$ of

Indeed, with $L_\gamma f=n\mathrm{dd}^{\mathrm{c}}f\wedge \gamma ^{n-1}/\gamma ^ n$ as in (11.23.1), we have $P_\gamma =2\pi L_\gamma $, and the adjoint probability of $L_\gamma $ is $\gamma ^ n/\int _ X\gamma ^ n$.

The curvature transforms by

On $X\setminus Q$, the $\gamma $-trace of the expression in parentheses is at least $t|v|_ h^2\rho =t|v|_ h^2>0$ for $v\ne 0$, so it has a positive eigenvalue there.

Over $Q$, the unit sphere bundle $\{ |v|_ h=1\} $ is compact. The largest $\gamma $-eigenvalue of $\alpha _{h,v}$ is continuous and strictly positive on it, so it has a positive minimum $\delta $. Let $C$ be the maximum on $X$ of the operator norm of $\mathrm i\partial \bar\partial f$ relative to $\gamma $. For every unit vector over $Q$, the largest eigenvalue of the expression in parentheses in (11.26.5) is at least $\delta -tC$, which is positive for small $t>0$. Homogeneity in $v$ and the positive factor $\mathrm e^{-tf}$ prove the assertion on $Q$ as well. $\square$