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 $M$.










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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 $M$.










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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 $M$.










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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






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      Todor

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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.






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            We discussed this paper at mathoverflow.net/questions/287275
            – David E Speyer
            yesterday










          • @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
            – Alexandre Eremenko
            yesterday











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          1 Answer
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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.






          share|cite|improve this answer



















          • 1




            We discussed this paper at mathoverflow.net/questions/287275
            – David E Speyer
            yesterday










          • @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
            – Alexandre Eremenko
            yesterday















          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.






          share|cite|improve this answer



















          • 1




            We discussed this paper at mathoverflow.net/questions/287275
            – David E Speyer
            yesterday










          • @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
            – Alexandre Eremenko
            yesterday













          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.






          share|cite|improve this answer














          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.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday

























          answered yesterday









          Alexandre Eremenko

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          48.8k6136252








          • 1




            We discussed this paper at mathoverflow.net/questions/287275
            – David E Speyer
            yesterday










          • @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
            – Alexandre Eremenko
            yesterday














          • 1




            We discussed this paper at mathoverflow.net/questions/287275
            – David E Speyer
            yesterday










          • @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
            – Alexandre Eremenko
            yesterday








          1




          1




          We discussed this paper at mathoverflow.net/questions/287275
          – David E Speyer
          yesterday




          We discussed this paper at mathoverflow.net/questions/287275
          – David E Speyer
          yesterday












          @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
          – Alexandre Eremenko
          yesterday




          @David E Speyer: Thanks for reminding. I remembered that I learned about this paper from MO, but no exact context.
          – Alexandre Eremenko
          yesterday


















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