Proposition 10.21.1.8D773ECA9ED2 Let $F:X\to Y$ be a finite surjective holomorphic map between compact Kähler manifolds of dimension $n$, let $\omega _ Y$ be a Kähler form on $Y$, and suppose that the class $F^*[\omega _ Y]$ contains a Kähler form $\theta $. Then the full-mass solution in the background $\theta $ with measure $(F^*\omega _ Y)^ n$ has a smooth bounded potential with minimal singularities, and its current is the smooth semipositive form $F^*\omega _ Y$. Fix a smooth positive local volume form $\mathrm{d}V$. In a chart at a generic point of the ramification divisor, away from its other components, in which $F(z,w)=(z^ k,w)$ with $k\geq 2$,
EA1615732A9D
10.21.2
\begin{equation} (F^*\omega _ Y)^ n=|z|^{2(k-1)}G\, \mathrm{d} V, \qquad F^*\omega _ Y\asymp |z|^{2(k-1)}|\mathrm{d} z|^2+|\mathrm{d} w|^2, \end{equation}
where $G$ is smooth and positive and $\asymp $ means that each side is bounded by a constant multiple of the other. The distance for $F^*\omega _ Y$ to the ramification divisor $\{ z=0\} $ is comparable to $|z|^ k$. Thus the metric on the complement of the ramification divisor is incomplete, although the current has no divisorial component. Finally, there is a strict barrier with logarithmic pole along the global ramification divisor.
Proof.
By the $\partial \bar\partial $-lemma on the compact Kähler manifold $X$, the de Rham class $F^*[\omega _ Y]$ determines the Bott–Chern class, as in subsection 20.4.9. Thus $F^*\omega _ Y=\theta +\mathrm{dd}^{\mathrm{c}}u$ for a smooth function $u$ on $X$. Its mass is the full mass of the class, and uniqueness of full-mass solutions in the closed theory, discussed in section 25.21, identifies $u$ with the solution up to a constant. The Jacobian determinant of $(z,w)\mapsto (z^ k,w)$ is $kz^{k-1}$, and comparison of $\omega _ Y$ with the Euclidean form gives (10.21.2). Hence every path to $\{ z=0\} $ has length bounded below by the variation of a fixed multiple of $|z|^ k$ along it, and a radial path with fixed $w$ attains this order.
The determinant of the differential $\mathrm{d}F$ defines the global ramification divisor $R$. Choose a smooth metric on $\mathcal O(R)$, with curvature form $\Theta _ R$ and canonical section $s_ R$. Since $\theta $ is Kähler, $\theta -\epsilon \Theta _ R>0$ for sufficiently small $\epsilon >0$, so by the Poincaré–Lelong formula $\epsilon \log |s_ R|^2-C$ is the potential of a strict positive $\theta $-current. The barrier is built from the global divisor $R$; no local branch coordinate is treated as a global line bundle.
$\square$
Pome
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