In order to carry out the present argument, which in essence is just a change of variables $z\rightsquigarrow\zeta=z^{1/m}$, we need to use analysis on certain smooth orbifolds. We will not give complete definitions since they are in the recent reference [BGK], section 2 pp. 560-564, see also [MO] and the references therein.

Let $(X,\Delta)$ be a smooth orbifold pair. By this we mean that we have the prime decomposition $\Delta=\sum_{i}(1-m_{i}^{-1})E_{i}$ where $m_{i}\in\mathbb{N}^{*}$ is an integer. We assume that $supp(\Delta)$ is a simple normal crossing divisor. Then, a classical construction surveyed in [BGK] enables to construct an orbifold $[X,\Delta]$ with a $1$-morphism of orbifolds $c:[X,\Delta]\to X$ with the following properties:

• •

  $c$ is the reduction to the coarse moduli space of $[X,\Delta]$.

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  $c_{X-Supp(\Delta)}:U:=[X,\Delta]\times_{X}\to X-Supp(\Delta)$ is an isomorphism. Hence $U$ is an open suborbifold of $[X,\Delta]$ which is an old-fashioned manifold).

• •

  For every open polydisk $\mathbb{D}\subset X$ with local coordinates $z_{1},\ldots,z_{n}$ such that $Supp(\Delta)=\{\prod_{j=1}^{p}z_{j}=0\}$ $[X,\Delta]\times_{X}\mathbb{D}=[\mathbb{D}^{\prime}/G_{loc}]$.

  In this formula, the local isotropy group is $G_{loc}=\prod_{j=1}^{p}\mathbb{Z}/m_{j}\mathbb{Z}$, $m_{j}$ is the integer multiplicity of the divisor $E_{i_{j}}$ such that $E_{i_{j}}\cap\mathbb{D}=\{z_{j}=0\}$, $G_{loc}$ acts on the polydisk $\mathbb{D}^{\prime}$ by $(\zeta_{1},...,\zeta_{p}).(z^{\prime}_{1},...,z^{\prime}_{n})=(\zeta_{1}z^{\prime}_{1},...,\zeta_{p}z^{\prime}_{p},z_{p+1}^{\prime},...)$ ⁹⁹ 9 The usual isomorphism of $\mathbb{Z}/m\mathbb{Z}$ with the group of $m$-th root of unity is used..

  The orbifold $1$-morphism $[\mathbb{D}^{\prime}/G_{loc}]\to\mathbb{D}$ is induced by $\varkappa_{loc}:(z^{\prime}_{1},..,z^{\prime}_{n})\mapsto((z^{\prime}_{1})^{m_{1}},...)$.

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  For sufficiently divisible $s$, $c_{*}\mathcal{O}_{[X,\Delta]}(sK_{[X,\Delta]})=\mathcal{O}_{X}(s(K_{X}+\Delta))$.

It is possible to define all the basic concepts of Kähler geometry on orbifolds such as smooth functions, Kähler metrics, etc… The principle is to think of $\varkappa_{loc}^{-1}$ as a (multivalued) smooth coordinate chart.

A continuous function on $[X,\Delta]$ is a continuous function on $X$. A Radon measure on $[X,\Delta]$ is a Radon measure on $X$.

A smooth function $f$ on $[X,\Delta]$ is a continuous function on $X$ such that for every local chart $\varkappa_{loc}^{*}f$ is smooth. In particular $f$ is Hölder continuous.

A Kähler metric $\Omega_{[X,\Delta]}$ on $[X,\Delta]$ is a Kähler metric $\Omega_{X-Supp(\Delta)}$ on $X-Supp(\Delta)$ with the property that $\varkappa_{loc}^{*}\Omega$ extends to a smooth Kähler metric on $\mathbb{D}^{\prime}$. In particular, it also extends as a closed Kähler current on $X$ with Hölder potentials.

The pull back of a Kähler form on $X$ to $[X,\Delta]$ is a semi-Kähler form that is actually cohomologous to a Kähler class¹⁰¹⁰ 10 Here no reference can be given. But it is easy to extend the gluing methods for Kähler forms developed in [Dem 3] and [Pa] to orbifolds. Hence [DP] extends to Kähler orbifolds. .

Observe that $\varkappa^{*}_{loc}dz_{l}=m_{i_{l}}(z^{\prime}_{l})^{m_{i_{l}}-1}dz^{\prime}_{l}$, hence a smooth volume form on $[X,\Delta]$ can be interpreted as a volume form $v$ on $X-Supp(\Delta)$ such that

$$v\text{ is comparable to }\frac{{\prod_{l=1}^{n}(\sqrt{-1}dz_{l}\wedge d\bar{z}_{l})}}{\prod_{l=1}^{p}|z_{l}|^{2(1-1/m_{i_{l}})}}.$$

In case the pair $(X,l^{-1}(t_{1}))$ is an orbifold pair, the equation

(5)

$$(\omega_{o}+dd^{c}\varphi_{t})^{n}=C_{t}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}}e^{F}\Omega_{X}^{n}$$

can be interpreted on $[X,\Delta]$ as an equation of the form

$$(c^{*}\omega_{o}+dd^{c}\varphi_{t})^{n}=C_{t}(|s_{1}|^{2k}+\ldots+|s_{p}|^{2k})e^{F}\Omega_{[X,\Delta]}^{n}.$$

The method used to analyze the case where $\cap_{i}\{t_{i}=0\}=\emptyset$ extends with almost no changes to the orbifold case. Hence the unique continuous solution of equation (5) is smooth outside its singular locus if $(X,\frac{1}{l}(t_{1}))$ is an orbifold pair.

Under the more general hypothesis that $\int_{X}|t_{1}|^{-2l}<\infty$ and $\frac{1}{l}(t_{1})$ is a divisor with simple normal crossings, then we can construct an orbifold pair $(X,\Delta)$ with $0\leq\frac{1}{l}(t_{1})\leq\Delta$ and we are back to the previous case.

For the most general case, consider the ideal $\mathcal{I}$ generated by the $t_{i}$ and fix $\mu:X^{\prime}\to X$ a log resolution of $(X,\mathcal{I})$. Then we are back to the previous case, with an equation on $X^{\prime}$ . This ends the proof of Theorem 3.6.

In certain rare circumstances, there is a finite smooth covering $Y\to X$ such that $Y/G=X$ and $[Y/G]=[X,\Delta]$ and the argument we use here reduces to a $G$-equivariant argument on $Y$.