Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb Z_m$ along $D$. Suppose now we have a Kähler orbifold metric $\omega$ on $(S,D,m)$ (with metric singularity along $D$). I would like to find a short proof or a reference for the following statement:

**Statement.** For arbitrary small $\varepsilon$-neighborhood $U_{\varepsilon}$ of $D$ in $S$ there is a smooth Kähler metric $\omega_{\varepsilon}$ on $S$ such that $\omega_{\varepsilon}$ coincides with $\omega$ in $S\setminus U_{\varepsilon}$.

**PS.** I will be happy, if the statement is proven in the following situation: $S$ is a ruled surface and $D$ is embedded in $S$ as a holomorphic section (i.e. it intersects all $\mathbb CP^1$-fibers in one point). Moreover if necessary one may assume that the Kähler orbi-metric is invariant under a holomorphic $S^1$-action on $S$ fixing $D$ pointwise.

**PPS.** I also removed the condition on $\omega_{\varepsilon}$ to be in the same cohomology class as $\omega$ (since it turned out to be irrelevant for me).

7more comments