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What is the “analytic†analogue of the valuative criterion of properness
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Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $mathbbC$. Let $D^*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion of properness in complex analysis.
Is the following correct?
The complex analytic space $X$ is compact if every holomorphic map $D^*to X$ extends to a holomorphic map $Dto X$.
The converse implication is not true, because there are non-extendable maps from $D^*$ to $mathbbP^1$, e.g., $zmapsto exp(-1/z^2)$.
I am thinking of $D^*$ as Spec $K$ and $D $ as Spec $R$, where $R$ is a dvr with fraction field $K$.
ag.algebraic-geometry complex-geometry
asked 2 hours ago
Sjoerd
432
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up vote
6
down vote
favorite
Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $mathbbC$. Let $D^*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion of properness in complex analysis.
Is the following correct?
The complex analytic space $X$ is compact if every holomorphic map $D^*to X$ extends to a holomorphic map $Dto X$.
The converse implication is not true, because there are non-extendable maps from $D^*$ to $mathbbP^1$, e.g., $zmapsto exp(-1/z^2)$.
I am thinking of $D^*$ as Spec $K$ and $D $ as Spec $R$, where $R$ is a dvr with fraction field $K$.
ag.algebraic-geometry complex-geometry
asked 2 hours ago
Sjoerd
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $mathbbC$. Let $D^*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion of properness in complex analysis.
Is the following correct?
The complex analytic space $X$ is compact if every holomorphic map $D^*to X$ extends to a holomorphic map $Dto X$.
The converse implication is not true, because there are non-extendable maps from $D^*$ to $mathbbP^1$, e.g., $zmapsto exp(-1/z^2)$.
I am thinking of $D^*$ as Spec $K$ and $D $ as Spec $R$, where $R$ is a dvr with fraction field $K$.
ag.algebraic-geometry complex-geometry
asked 2 hours ago
Sjoerd
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $mathbbC$. Let $D^*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion of properness in complex analysis.
Is the following correct?
The complex analytic space $X$ is compact if every holomorphic map $D^*to X$ extends to a holomorphic map $Dto X$.
The converse implication is not true, because there are non-extendable maps from $D^*$ to $mathbbP^1$, e.g., $zmapsto exp(-1/z^2)$.
I am thinking of $D^*$ as Spec $K$ and $D $ as Spec $R$, where $R$ is a dvr with fraction field $K$.
ag.algebraic-geometry complex-geometry
ag.algebraic-geometry complex-geometry
asked 2 hours ago
Sjoerd
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Sjoerd
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Sjoerd
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Sjoerd
432
asked 2 hours ago
Sjoerd
432
432
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Sjoerd is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
add a comment |Â
1 Answer
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oldest
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up vote
4
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No, the open disk in $mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
answered 1 hour ago
John Pardon
8,934230102
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
No, the open disk in $mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
answered 1 hour ago
John Pardon
8,934230102
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
add a comment |Â
up vote
4
down vote
accepted
No, the open disk in $mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
answered 1 hour ago
John Pardon
8,934230102
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
No, the open disk in $mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
answered 1 hour ago
John Pardon
8,934230102
No, the open disk in $mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
answered 1 hour ago
John Pardon
8,934230102
answered 1 hour ago
John Pardon
8,934230102
answered 1 hour ago
John Pardon
8,934230102
answered 1 hour ago
John Pardon
8,934230102
8,934230102
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
add a comment |Â
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
Ok. That's of course correct. But what if I assume $X$ is quasi-projective in addition?
– Sjoerd
27 mins ago
add a comment |Â
Sjoerd is a new contributor. Be nice, and check out our Code of Conduct.
Sjoerd is a new contributor. Be nice, and check out our Code of Conduct.
Sjoerd is a new contributor. Be nice, and check out our Code of Conduct.
Sjoerd is a new contributor. Be nice, and check out our Code of Conduct.
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