Let $V$ be a normal analytic space of pure dimension $n$. A plurisubharmonic (psh) function $\varphi$ on $V$ is an upper semicontinuous function on $V$ with values in $\mathbb{R}\cup\{-\infty\}$, which is not locally $-\infty$, and extends to a psh function in some local embedding $V\to\mathbb{C}^{N}$. The function $\varphi$ is strongly psh (resp. ${\mathcal{C}}^{0}$, resp. ${\mathcal{C}}^{\infty}$) iff it extends to a strongly psh function (resp. ${\mathcal{C}}^{0}$, resp. ${\mathcal{C}}^{\infty}$) in some local embedding. A continuous function is psh iff its restriction to $V^{reg}$ is so [FN]. A bounded psh function on $V^{reg}$ extends to $V$.

A pluriharmonic function on $V$ is a real valued continuous function on $V$ $f$ on $V$ such that one of the following equivalent conditions holds:

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  $f$ is locally the real part of a holomorphic function.

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  Given a local embedding $V\to\mathbb{C}^{N}$, $f$ extends locally to a pluriharmonic function on $\mathbb{C}^{N}$.

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  $f|_{V^{reg}}$ is pluriharmonic.

Define an equivalence relation on semi-kähler potentials requiring that $(U_{i},\varphi_{i})\sim(V_{j},\psi_{j})$ iff $\varphi_{i}-\psi_{j}$ is pluriharmonic on $U_{i}\cap V_{j}$.

We will on occasion drop the requirement that the local potentials of $\Omega$ are psh, replacing it by the requirement that they are locally the sum of a smooth and a psh function. The current $\Omega$ will then be called a quasi positive closed current on $V$.

If it has locally bounded potentials, $\Omega$ is fully determined by the closed $(1,1)$ form $\Omega_{reg}$ on $V_{reg}$ defined on $U_{i}$ by $\Omega_{reg}=dd^{c}\varphi_{i}$.

Let $\Omega$ be a smooth Kähler metric on $V$ with Kähler potential $(U_{i},\varphi_{i})$. An upper semi-continuous function $\varphi:X\to\mathbb{R}\cup{-\infty}$ is said to be $\Omega$-psh iff $\forall i$ $\varphi_{i}+\varphi$ is psh on $U_{i}$. The semi-Kähler current whose potential is $(U_{i},\varphi+\varphi_{i})$ is denoted by $\Omega+dd^{c}\varphi$.

The metric $dd^{c}\varphi^{2}$ is an example of an orbifold Kähler metric on $V$. The results of [Y] extend without major modifications to Kähler orbifolds. For instance, in each Kähler class of a nodal K3 surface there is a unique Ricci flat orbifold metric.

Let $\mathcal{PH}_{V}$ be the sheaf of real-valued pluriharmonic functions on $V$. By definition, a closed (1,1)-form on $V$ is a section of the sheaf ${\mathcal{C}}^{\infty}_{V}/\mathcal{PH}_{V}$. We have the exact sequence:

A class in $H^{1}(X,\mathcal{PH}_{X})$ will be called Kähler, if it is in the $[\ \ .\ \ ]$ image of a smooth Kähler metric.

Let $L$ be a holomorphic line bundle on $V$. The notion of smooth hermitian metric on $(V,L)$ is defined as in the smooth case. Let $h$ be such a metric on $(V,L)$.

Let $s\in H^{0}(U,L)$ be a nowhere zero local holomorphic section of $L$ (a local generator of $L$) defined over the open subset $U\subset V$. Set $e^{-\varphi_{s}}:=||s||_{h}^{2}$, where $\varphi_{s}$ is a ${\mathcal{C}}^{\infty}$-smooth function on $U$. The current $dd^{c}\varphi_{s}$ is a smooth closed (1,1)-form on $V$ which does not depend on $s$; it is a semi-Kähler current if $\varphi_{s}$ is psh.

More generally, let $(U_{i})_{i}$ be an open covering of $V$ and $s_{i}\in H^{0}(U_{i},\mathcal{O}_{V}(L))$ a local generator of $L$. Let $\varphi_{i}=\varphi_{s_{i}}$. The datum $(U_{i},\varphi_{i})$ defines a smooth closed (1,1)-form on $V$.

It is immediate that $[c_{1}(L,h)]$ is independent of $h$. Hence there is a linear map $c_{1}:Pic(V)\to H^{1}(V,\mathcal{PH}_{V})$. The connection with the more widely known smooth case is made by the observation that, if $X$ is a compact Kähler manifold, $H^{1,1}(X,\mathbb{R})=H^{1}(X,\mathcal{PH}_{X})$.

A singular metric on $L$ is an expression $h=e^{-\varphi}h_{sm}$, $\varphi$ being a locally smooth + psh function and $h_{sm}$ a smooth hermitian metric. Its Chern-Weil form is the quasi-positive current $c_{1}(L,h_{sm})+dd^{c}\varphi$.