Smoothing of a Kähler orbifold metric on a complex surface

Question

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).

Answer 1

Let $S=S^{sing}\cup S^{reg}$ take an orbifold resolution $\pi : \tilde S \to S$ with simple normal crossing exceptional divisor $E = \pi^{−1}(S^{sing})$ such that $\pi$ is an isomorphism over $S^{reg}$. then take $\omega_\epsilon=\pi^*\omega_0-\epsilon\theta_E$ as K\"ahler metric on $\tilde S $ in adiabatic classes(see Lemma 4.2.3 that why such metric $\omega_\epsilon$ is as Kahler metric in fibration case or Lemma1 for resolution case . Here $\omega_0$ is a positive closed $(1,1)$-current on $S$ and $\theta_E$ is a positive closed $(1,1)$-current on $E$)

Consider a smooth geometric orbifold given by $\mathbb Q$-divisor $$D=\sum_{j\in J}(1-\frac{1}{m_j})D_j$$ where $m_j\geq 2$ are positive integers and $\text{Supp}D=\cap_{j\in J}D_j$ is of normal crossings divisor. Let $\omega$ be any K\"ahler metric on $S$, let $C >0$ be a real number and $s_j\in H^0\left(S,\mathcal O_X(D_j)\right)$ be a section defining $D_j$. Consider the following expression

$$\omega_D=C\omega+\sqrt[]{-1}\sum_{j\in J}\partial\bar\partial |s_j|^{2/m_j}$$

If $C$ is large enough, the above formula defines a closed positive $(1,1)$ -current (smooth away from $D$). Moreover

$$\omega_D\geq \omega$$ in the sense of currents. Consider $\mathbb C^n$ with the orbifold divisor given by the equation

$$\prod_{j=1}^nz_j^{1-{1}/{m_j}}=0$$ (with eventually $m_j=1$ for some $j$). The sections $s_j$ are simply the coordinates $z_j$ and a simple computation gives

$$\omega_D=\omega_{eucl}+\sqrt[]{-1}\sum_{j=1}^n\partial\bar\partial |z_j|^{2/m_j}=\omega_{eucl}+\sqrt[]{-1}\sum_{j=1}^n\frac{dz_j\wedge d\bar {z_j}}{m_j^2|z_j|^{2(1-1/m_j)}}$$

See paper of Frédéric Campana; Mihai Păun Orbifold generic semi-positivity: an application to families of canonically polarized manifolds (Semi-positivité orbifolde : une application aux familles de variétés canoniquement polarisées) Vol. 65 no. 2 (2015), p. 835-861 Details

http://www.numdam.org/article/AIF_2015__65_2_835_0.pdf