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.

  • $\begingroup$ 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. $\endgroup$ 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. $\endgroup$ Feb 7 '18 at 4:32

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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.