Kähler metric on compact complex manifolds with simple normal crossing divisor

Question

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

Answer 1

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 (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