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Can one explain what is differential form on a Riemann Surface? What is motivation to define it?

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Do you know what a differential form on a smooth manifold is? – Qiaochu Yuan May 18 '11 at 6:09
For one, differential forms provide a coordinate-invariant description of things to integrate over a Riemann surface. Unless you want to die a slow, painful death in notation bookkeeping, differential forms are your friend. – Gunnar Þór Magnússon May 18 '11 at 7:04
There is an analytic chart-based definition of meromorphic differential on page 25 of these notes, but there are other approaches (especially on compact Riemann surfaces, which are algebraic curves) and other kinds of differentials (e.g. $C^\infty$). – Zhen Lin May 18 '11 at 8:41

On a compact Riemann surface, any holomorphic function is constant (Liouville's theorem). So one looks for alternatives. Meromorphic functions are plentiful (though it takes some hard analysis to prove that!) and helpful, especially for generating maps from the Riemann surface to projective space. These mapping illuminate the geometry of the Riemann surface. For example, you can look at a surface as a branched cover of the Riemann sphere.

You can play the same trick with differential forms, for example, generating a morphism to an object called the Jacobian variety.

Finally, via something called Serre Duality and the Riemann Roch theorem, you get an important relationship between the sections of line bundles associated with a divisor and meromorphic differential forms associated with the negative of the same divisor.

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