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Bumped by Community user
Bumped by Community user
Bumped by Community user
Removed “(local meromorphic)” after the definition of s(z)
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Kwok Kin Wong
Kwok Kin Wong
  • 87
  • 2

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 (local meromorphic) 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.

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 (local meromorphic) 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.

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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Kwok Kin Wong
Kwok Kin Wong
  • 87
  • 2

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 (local meromorphic) 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.