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Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that method ad hoc and I would like a more natural proof.)

Note: by "positive", I mean a line bundle equipped with a Hermitian metric of positive curvature form.

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    $\begingroup$ A way to show that an ample line bundle has a positive metric avoiding Hodge theory is to use Bergman kernel asymptotics. See for example Theorem 6.7 here: homepages.vub.ac.be/~joelfine/preprints/2012-07_Cologne.pdf The proof is quite hands-on. $\endgroup$ – Ruadhaí Dervan Jan 18 '15 at 13:42
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    $\begingroup$ You can find a direct construction of positive line bundles on a compact Riemann surface on page 160 of the book by Napier and myself An Introduction to Riemann surfaces .You can also look at the paper in L'Enseignment Mathematique vol 50 pages 367-390 $\endgroup$ – Mohan Ramachandran Jan 18 '15 at 17:02
  • $\begingroup$ thanks Prof.Mohan very much.I am actually preparing to give a talk on embedding of compact riemann surface in complex projective space using hormander's method from varolin's book.there are various subtle points not fully explained in the book,your book will greatly help me in that. $\endgroup$ – Koushik Jan 18 '15 at 17:31
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Denote this Riemann surface by $X$. It is equivalent to find a complex line bundle of positive degree on $X$.

We can identify the isomorphism classes of line bundles on $X$ with $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ via Cech data, where $\mathcal{C}^{\infty}_X(\mathbb{C}^*)$ is the sheaf of smooth $\mathbb{C}^*$-valued functions on $X$. Furthermore, the exponential sequence $$0\rightarrow \underline{\mathbb{Z}}\rightarrow\mathcal{C}^{\infty}_X(\mathbb{C})\rightarrow\mathcal{C}^{\infty}_X(\mathbb{C}^*)\rightarrow 0$$ yields a long-exact sequence in sheaf cohomology that identifies $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ with $H^2(X;\underline{\mathbb{Z}})\cong\mathbb{Z}$. This identification associates to a line bundle in $H^1(X;\mathcal{C}^{\infty}_X(\mathbb{C}^*))$ its degree in $\mathbb{Z}$,

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  • $\begingroup$ I think you are missing the point. "Positive" means ample, that is, the sections of a sufficiently high multiple of your line bundle embeds $X$ into some projective space. $\endgroup$ – abx Jan 17 '15 at 16:44
  • $\begingroup$ "positivity has various meanings but what I mean is a line equipped with a hermitian metric of positive curvature form. $\endgroup$ – Koushik Jan 17 '15 at 16:49
  • $\begingroup$ Okay, it's clear that I misinterpreted the question. $\endgroup$ – Peter Crooks Jan 17 '15 at 17:26

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