Let $X$ be a reduced compact complex analytic space of $\dim\_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a *strong resolution* $R(X)$ which is smooth, $E=\pi\_X^{-1}(X\_{sing})$ is a *simple normal crossing (snc) divisor*, $R(X)\setminus E$ is biholomorphic to $X\_{reg}$ and the blow down morphism $\pi\_X$ is projective; in other words $R(X)$ is a compact complex manifold with a snc divisor $E$.


I recall the definition of Kähler metric $g$ on $X$ in the Grauert's sense ([GH] §3.3).



> 
> Let $g$ be a Hermitian metric on $X\_{reg}$. $g$ is a *Kähler metric on $X$* if there exist an open covering $\{U\_l\}\_{l\in\{1,\dots,m\}}$ of $X$ and *strongly plurisubharmonic functions* (strongly psh) $p\_l$ defined on $U\_l$ such that on $U\_l\setminus X\_{sing}$ one has $\displaystyle g\_{h\overline{k}|U\_l}=\frac{\partial^2p\_l}{\partial z^h\overline{\partial}z^k}$; where $z^1,\dots,z^n$ are the local coordinates of $X$ on $U\_l$. $p\_l$ is called *local Kähler potential for $g$ on $U\_l$*.
> 
> 
> 


If $X$ admits a Kähler metric $g$, I can pull-back $g$ on $R(X)\setminus E$ via $\pi\_X$.


**Questions.** Can I extend or "deform" $\pi\_X^{\*}g$ on the whole of $R(X)$ in order to define a $C^2$-Hermitian metric on $R(X)$?




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[GH] H. Grauert - *Über Modifikationen und exzpetionelle analytische Mengen*,
Math. Annalen **146** (1962) 331-368


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



Kähler metric on compact complex manifolds with simple normal crossing divisor
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:


1. $\widehat{X}$ is a normal complex analytic space;
2. $\nu$ is a finite surjective holomorphic map;
3. $X\_{norm}$ is the set of normal points of $X$, and its complement is [*thin*](https://en.wikipedia.org/wiki/Thin_set_(analysis)) (cfr. [GH,RR] statement 6.2.2);
4. $\nu^{-1}(X\setminus X\_{norm})$ is thin;
5. $\nu:\widehat{X}\setminus\nu^{-1}(X\setminus X\_{norm})\to X\_{norm}$ 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 $\beta\_X^{\*}g$.




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[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


