Theorem 11.31.2160CE026081B

Proof. The set $\mathcal K^0_{\theta ,\mathrm{d}V}$ is convex. It is the image of the compact set $\mathcal K_\theta $ of theorem 5.25.2 under the continuous map $v\mapsto v-\int _ Xv\, \mathrm{d}V$, and is therefore compact.

If $u_ j\to u$ in $L^1$, then $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}u$ as currents, so the trace measures converge in the sense of distributions. Since they are positive measures of the same finite mass, and every continuous function is a uniform limit of smooth functions, they also converge weakly as measures. Thus $\mathcal T_\gamma $ is continuous.

To prove injectivity, suppose that $\mathcal T_\gamma (u)=\mathcal T_\gamma (v)$. Then $A(u-v)=0$ in the sense of distributions. Elliptic regularity for $A$ shows that $u-v$ agrees almost everywhere with a smooth function, and the maximum principle shows that this function is constant. Its mean is zero, so $u=v$ almost everywhere, and hence everywhere by lemma 5.24.3.

A continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image; here the target is the space of positive measures of mass $M_{\theta ,\gamma }$ with the weak topology. Since $\mathcal T_\gamma $ is affine, its image is convex. If $M_{\theta ,\gamma }=0$, every trace measure is zero, so injectivity shows that $\mathcal K^0_{\theta ,\mathrm{d}V}$ has at most one point; it has exactly one when $\mathrm{PSH}(X,\theta )$ is nonempty, since the mean of any $\theta $-plurisubharmonic function can be subtracted. In this case lemma 11.6.3 also gives $\theta +\mathrm{dd}^{\mathrm{c}}u=0$ for the unique normalized potential $u$, which is therefore smooth. $\square$