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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      asked 3 hours ago









      Todor

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          Such $f$ exists on any open Riemann surface:



          R.C. Gunning and R. Narasimhan. Immersion of open Riemann surfaces.
          Math. Ann., 174:103–108, 1967.






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            1 Answer
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            1 Answer
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            active

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            up vote
            3
            down vote













            Such $f$ exists on any 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

























              up vote
              3
              down vote













              Such $f$ exists on any 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























                up vote
                3
                down vote










                up vote
                3
                down vote









                Such $f$ exists on any 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 any 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



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                answered 2 hours ago









                Alexandre Eremenko

                48.5k6134250




                48.5k6134250






























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