The cotangent bundle of a non-compact Riemann surface
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Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a holomorphic 1-form $omega$ that does not have any zeroes on $M$. I am interested to know whether we can find an exact 1-form $omega=df$, where $f$ is some holomorphic function on $M$, such that $omega$ does not have zeroes on $M$.
ag.algebraic-geometry
asked Dec 5 at 3:34
Todor
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up vote
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Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a holomorphic 1-form $omega$ that does not have any zeroes on $M$. I am interested to know whether we can find an exact 1-form $omega=df$, where $f$ is some holomorphic function on $M$, such that $omega$ does not have zeroes on $M$.
ag.algebraic-geometry
asked Dec 5 at 3:34
Todor
561
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a holomorphic 1-form $omega$ that does not have any zeroes on $M$. I am interested to know whether we can find an exact 1-form $omega=df$, where $f$ is some holomorphic function on $M$, such that $omega$ does not have zeroes on $M$.
ag.algebraic-geometry
asked Dec 5 at 3:34
Todor
561
Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a holomorphic 1-form $omega$ that does not have any zeroes on $M$. I am interested to know whether we can find an exact 1-form $omega=df$, where $f$ is some holomorphic function on $M$, such that $omega$ does not have zeroes on $M$.
ag.algebraic-geometry
ag.algebraic-geometry
asked Dec 5 at 3:34
Todor
561
asked Dec 5 at 3:34
Todor
561
asked Dec 5 at 3:34
Todor
561
asked Dec 5 at 3:34
Todor
561
asked Dec 5 at 3:34
Todor
561
561
add a comment |
add a comment |
1 Answer
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Such $f$ exists on every open Riemann surface:
R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
Math. Ann., 174:103–108, 1967.
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
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up vote
10
down vote
Such $f$ exists on every open Riemann surface:
R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
Math. Ann., 174:103–108, 1967.
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
add a comment |
up vote
10
down vote
Such $f$ exists on every open Riemann surface:
R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
Math. Ann., 174:103–108, 1967.
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
add a comment |
up vote
10
down vote
up vote
10
down vote
Such $f$ exists on every open Riemann surface:
R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
Math. Ann., 174:103–108, 1967.
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
Such $f$ exists on every open Riemann surface:
R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
Math. Ann., 174:103–108, 1967.
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
edited Dec 5 at 18:48
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
answered Dec 5 at 4:18
Alexandre Eremenko
48.9k6136253
48.9k6136253
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
add a comment |
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
1
1
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
We discussed this paper at mathoverflow.net/questions/287275
– David E Speyer
Dec 5 at 19:30
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
@David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
– Alexandre Eremenko
Dec 6 at 0:13
add a comment |
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