Proposition 10.25.1D05A6736BCFF

Proof. In a coordinate neighborhood meeting $D$, write $s_ D=z_1 e$, where $e$ is a local holomorphic frame with $|e|^2=\mathrm{e}^{-\psi }$ and $\psi $ is smooth. Then

Covering the compact divisor by finitely many such neighborhoods and comparing with the smooth background, we obtain, after increasing $\tau _0$,

for positive constants $c_1,C_1$. In particular, $E_\tau $ is a hyperplane there.

At a point of this set, choose an $\omega $-unitary frame whose last $n-1$ vectors span $E_\tau $. Let $H$ be the Hermitian matrix that represents $g_ T$ as the quadratic form $w^*Hw$ in the complex components $w$, and write

where $a$ is a column vector and $a^*$ its conjugate transpose. Completing the square gives, for a vector with components $(\xi ,\eta )$,

For $n=1$ the same identity holds without the second term. For the complex components of a real vector $v$ we have $|\xi |=|\partial \tau (v)|/|\partial \tau |_\omega $, so (10.25.2) and (10.25.3) imply

where $c_2>0$ is uniform. The last inequality uses $\mathrm{d}\tau (v)=2\operatorname {Re}\partial \tau (v)$.

Define a continuous function $\Phi _ p$ on $[1,\infty )$ by

Integrating (10.25.4) along piecewise smooth curves shows that $\Phi _ p\circ \tau $ is Lipschitz for the length distance of $g_ T$; this also holds for curves crossing $\{ \tau =\tau _0\} $, since $\Phi _ p$ is locally Lipschitz and constant below that level. Since $p\leq 2$, $\Phi _ p(t)$ tends to infinity with $t$, so $\Phi _ p\circ \tau $ is a proper function on $M$.

A Cauchy sequence for the length distance has bounded values of this Lipschitz function and hence lies in a compact subset of $M$. It therefore has a subsequence converging in the manifold topology. On a coordinate ball about the limit, the smooth positive form $T$ is comparable to the Euclidean metric, so this subsequence also converges for the length distance. The Cauchy property then gives convergence of the whole sequence, which proves completeness. $\square$