Plurisubharmonic function having log pole along divisor 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.
 A: 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".
