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