Without loss of generality, let $X$ be irreducible, then it is pure-dimensional ([GH,RR] proposition 9.1.3).

Let $\left(\widehat{X},\nu\right)$ be the normalization of $X$ ([FG] chapter 2, section 16, Normalization theorem); for exact:

- $\widehat{X}$ is a normal complex analytic space;
- $\nu$ is a finite surjective holomorphic map;
- $X_{norm}$ is the set of normal points of $X$, and its complement is
*thin* (cfr. [GH,RR] statement 6.2.2);
- $\nu^{-1}(X\setminus X_{norm})$ is thin;
- $\nu:\widehat{X}\setminus\nu^{-1}(X\setminus X_{norm})\to X_{norm}$ is a biholomorphism.

Because $ X_{norm}\supseteq X_{reg}$, one pull-backs $g$ on the open subset $\nu^{-1}(X_{reg})=A$ of $\widehat{X}$; by construction:

- $\nu^{-1}(X_{sing})$ is a thin subset of $\widehat{X}$: by [GH,RR] statement 6.2.2, $X_{sing}$ is thin and by finiteness of $\nu$ follows the "thinness";
- $A=\dots=\widehat{X}\setminus\nu^{-1}(X_{sing})$.

By [HL] theorem 6.2.4, in according to [VJ.1] section 1 (see also [RR]), $\nu^{*}p_l=\widehat{p_l}$ are local Kähler potentials of $\nu^{*}g_{|A}=\widehat{g}_{|A}$.

Without loss of generality, one can assume that $\widehat{p_l}=q_l+r_l$ where $q_l$ is strongly psh and $r_l$ is the real part of holomorphic map $f_l$ on $V_l=\nu^{-1}(U_l)\cap A$; then $\forall l_1,l_2\in\{1,\dots,m\},\,q_{l_1}-q_{l_2}=r_{l_{1,2}}$ is *pluriharmonic* (*ph*) on $V_{l_1}\cap V_{l_2}=V_{l_{1,2}}$.

By [GH,RR] corollary 7.4.2, any $f_l$ is extendible on the whole of $\nu^{-1}(U_l)$.

One calls $\widetilde{f}_l$ these holomorphic functions on $\nu^{-1}(U_l)$; in consequence, one has local Kähler potential $\widetilde{p_l}$ and let $\widetilde{g}$ be the relative Kähler metric on $\widehat{X}$. By construction $\widetilde{g}_{|\widehat{X}\setminus\nu^{-1}(X\setminus X_{norm})}=\nu^{*}g$, so $\widetilde{g}$ is an extension of $\nu^{*}g$ on the whole of $\widehat{X}$.

**Remark 1.** By [VJ.2] theorem 1: $\widehat{g}$ is also a Kähler metric on $\widehat{X}$ is the Moišhezon's sense. More in general, in this way I had prove that the Kähler metrics in the sense of Grauert and Moišhezon are equivalent over normal (irreducible) complex analytic spaces.

Let $\left(R(X),\beta_X\right)$ be the strong resolution of $X$ (cfr. [KF]); in particular, it is a normal (irreducible compact) complex analytic space and, by [GH,RR] proposition 8.4.3, one can consider the *normal lifting* $\widehat{\beta_X}:R(X)\to\widehat{X}$ of $\beta_X$.

**Remark 2.** By construction $\beta_X=\nu\circ\widehat{\beta_X}$.

Because $\nu$ is a finite surjective continuous map, then $\widehat{X}=\nu^{-1}(X)$ is a compact space; one can construct its strong resolution $\left(R\left(\widehat{X}\right),\beta_{\widehat{X}}\right)$ as complex analytic space and the *resolution morphisms* $\widetilde{\nu}:R\left(\widehat{X}\right)\to R(X)$ and $\gamma:R(X)\to R\left(\widehat{X}\right)$ of $\nu$ and $\widehat{\beta_X}$, respectively. (cfr. [KF]). By [BJ] lemma 4.4 and proposition 4.6.(4) (cfr. [VJ.2] proposition II.1.3.1.(V) and (VI)) $R\left(\widehat{X}\right)$ is a smooth Kähler space.

Because
\begin{equation}
\widehat{\beta_X}\circ\widetilde{\nu}\circ\gamma=\beta_{\widehat{X}}\circ\gamma=\widehat{\beta_X}\Rightarrow\widetilde{\nu}\circ\gamma=Id_{R(X)}\\
\beta_{\widehat{X}}\circ\gamma\circ\widetilde{\nu}=\widehat{\beta_X}\circ\widetilde{\nu}=\beta_{\widehat{X}}\Rightarrow\gamma\circ\widetilde{\nu}=Id_{R\left(\widehat{X}\right)}
\end{equation}
one has that $R(X)$ is biholomorphic to $R\left(\widehat{X}\right)$; in particular $R(X)$ has a Kähler metric $\left(\widehat{\beta_X}\right)^{*}\widetilde{g}$ such that its restriction to $R(X)\setminus\beta_X^{-1}\left(X_{sing}\right)\equiv R(X)\setminus E$ is $\beta_X^{*}g$.

[BJ] J. Bingener - *On Deformation of Kähler spaces. I*, Math. Z. **182** (1983) 505-535

[FG] G. Fischer (1976) *Complex Analytic Geometry*, Springer-Verlag

[GH,RR] H. Grauert, R. Remmert (1984) *Coherent Analytic Sheaves*, Springer-Verlag

[HL] L. Hörmander (1990) *An Introduction to Complex Analysis in Several Variables. Second edition*, North-Holland

[KJ] J.Kollár (2007) *Lectures on Resolutions of Singularities*, Princeton University Press

[RR] R. Richberg - *Stetige Streng Psedokonvexe Functionen*, Math. Ann. **175** (1968) 257-286

[VJ.1] J. Varouchas - *Sur l'image d'une variété kählérienne compacte*, Séminaire F. Norguet. Fonctions de Plusieurs Variables Complexes V. Lecture Notes in Mathematics, **1188** (1986) Springer-Verlag

[VJ.2] J. Varouchas - *Kähler Spaces and Proper Open Morphisms*, Math. Ann. **283** (1989) 13-52