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

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 $$R(X)$$?

[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

• It might be worth noting that the notion of "plurisubharmonic" function on a complex analytic space was a problem in several complex variables that was treated by Fornaess and Narasimhan, see their paper concerning the Levi problem on Stein spaces. – Very Confused Aug 19 '20 at 2:40
• Yes, indeed I consider (strongly) psh functions on $X_{reg}$: is not this enough? – Armando j18eos Aug 19 '20 at 6:52
• If I recall correctly, at least for reduced complex spaces, Fornaess and Narasimhan showed the different notions of PSH defined in their paper are all equivalent. – Very Confused Aug 19 '20 at 8:07

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 $$$$\widehat{\beta_X}\circ\widetilde{\nu}\circ\gamma=\beta_{\widehat{X}}\circ\gamma=\widehat{\beta_X}\Rightarrow\widetilde{\nu}\circ\gamma=Id_{R(X)}\\ \beta_{\widehat{X}}\circ\gamma\circ\widetilde{\nu}=\widehat{\beta_X}\circ\widetilde{\nu}=\beta_{\widehat{X}}\Rightarrow\gamma\circ\widetilde{\nu}=Id_{R\left(\widehat{X}\right)}$$$$ one has that $$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

• I have read through this proof, I'm not sure if I'm convinced that you could extend the Kähler metric over the codimension part. Maybe this is only a problem if you consider the singularities of a Kähler space together with the critical values of some map (the discriminant locus). – Very Confused Aug 19 '20 at 8:09
• For exact, in your technical opinion: what is the fake step in my proof? – Armando j18eos Aug 20 '20 at 10:15