Theorem 10.28.4195B9613F5AA

Proof. Suppose that such a metric exists. Since $\pi $ is proper, we may shrink $\Delta $ so that the whole family lies in its domain: otherwise points outside the neighborhood over $t_ j\to 0$ would have a limit point on $X_0$ outside it. The lower bound makes the restricted weight finite at every point of $q\setminus \{ p\} $, because $\varphi _0$ is smooth there. Hence the restrictions of the weight to $\Gamma $ and to $q$ are not identically $-\infty $.

Near a point of $q\setminus \{ p\} $, choose a product chart $(\zeta ,t)$ on $\Gamma $, as given by lemma 10.28.3, and a holomorphic frame of $L$. The restricted weight $u(\zeta ,t)$ is plurisubharmonic and locally integrable. By Fubini's theorem, for almost every small $t\ne 0$ its restriction to $C_ t$ is not identically $-\infty $, so the metric restricts to $C_ t$. The curvature of this restriction is a positive measure on the compact curve $C_ t$ of total mass $L_ t\cdot C_ t=0$, so it vanishes. Thus $u(\cdot ,t)$ is harmonic for almost every $t$. By lemma 10.28.6 below, whose proof does not use this theorem, the central slice $u(\cdot ,0)$ is harmonic or identically $-\infty $, and it is finite. Hence the restricted weight is harmonic near every point of $q\setminus \{ p\} $.

The restricted metric on $L_0|_ q$ therefore has curvature supported at $p$. This curvature is a positive measure of total mass $L_0\cdot q=1$, so it is the Dirac measure $[p]$. In a coordinate $w$ on $q$ centered at $p$, the restricted weight is therefore $\log |w|^2$ plus a harmonic function, because their difference has zero distributional Laplacian. Since $\ell $ and $q$ meet transversely, $\varphi _0|_ q=\tfrac 12\log |w|^2+O(1)$. The difference $\psi |_ q-\varphi _0|_ q$ thus tends to $-\infty $ at $p$, contradicting the assumed lower bound. $\square$