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Let $R$ be a compact Riemann surface. For a given point $p\in R$ identified to the origin $z=0$ in a coordinate chart, then the function $z$ defines a local holomorphic section vanishing along the divisor D=(p). We may take $$ \varphi=\log |z|^2 $$ then $$ s(z):=e^{-k\varphi}, \quad k\in \mathbb{N}, $$ is a function near $p$ and having a pole at $p$.

My question is, how far can this be generalized to higher dimensions?

For instance, it seems to me that at least this can be generalized to the case that $R=X$ is a projective manifold and $D\subset X$ is a smooth divisor. Say $D=\{z_1=0\}$ in a coordinate chart, then the construction of a (local) plurisubharmonic function having a log pole along $D$ goes as in the case of compact Riemannian surface, if I where not mistaken.

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  • $\begingroup$ You define $\phi$ to equal $\log |z|^2$, and you define $s$ to equal $e^{-k\phi}$. Thus, $s$ equals $|z|^{-2k}$. Why do you write that $s$ "is a (local meromorphic) function near $p$ and having a pole at $p$"? For a holomorphic coordinate $z$, the function $|z|^2$ is not a local meromorphic function. $\endgroup$ – Jason Starr Feb 6 '18 at 8:27
  • $\begingroup$ I guess all I want is that s(z) has a pole singularity at 0. Thanks for the clarification. $\endgroup$ – user120515 Feb 7 '18 at 4:21
  • $\begingroup$ s(z) is not meromorphic. I guess all I want is only that s(z) has pole singularity at 0. $\endgroup$ – Kwok Kin Wong Feb 7 '18 at 4:32
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It is not clear what you are asking (what exactly do you want to generalize?), but let's start with a definition. Plurisubharmonic functions with logarithmic singularities can and have been be studied in complex manifolds of dimensions higher than $1$. In MR2133260 Rashkovskii, Alexander Sigurdsson, Ragnar Green functions with singularities along complex spaces. Internat. J. Math. 16 (2005), no. 4, 333–355 the authors consider a complex subspace $A$ of a connected complex manifold $X$ (not necessarily of codimension one) and $\mathcal{J}_A=(\mathcal{J}_A,x)_{x\in X}$-- the coherent sheaf of ideals in $\mathcal{O}(X)$ associated to $A$. They define the class $\mathcal{F}_A$ as the set of all negative plurisubharmonic functions $u$ on $X$ such that for every point $a \in X$ there exist local generators $\psi_1,…,\psi_m$ for $\mathcal{J}_A$ near $a$ and a real constant $C$ (depending on $u$ and the generators) such that $u\leq \log\|\psi\|+C$ near $a$.

This cannot be applied to a compact $X$ (no non-constant plurisubharmonic functions!), but one still can develop somewhat analogous theory for so-called $\omega$-plurisubharmonic functions with respect to a closed real $(1,1)$-current $\omega$ on $X$ (when such functions exist). For the details I recommend MR2203165 Guedj, Vincent; Zeriahi, Ahmed Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15 (2005), no. 4, 607–639. Of particular interest to you should be the beginning of Section 6, where $[\omega]$ is a the first Chern class of a holomorphic line bundle $L$ over $X$. The observation that there is a 1-to-1 correspondence between $\omega$-plurisubharmonic functions and positive singular metrics on $L$ should help you make sense of ``plurisubharmonic functions with logarithmic singularities along a divisor".

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