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I am reading the draft of "Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups" and the author starts using the notation $H^0(C, mK)$ on page 3, without explaining it. As the author is studying the action of the automorphism group of $C$ on $H^0(C, mK)$ and refers to Farkas who studies the action of the automorphism group of $C$ on holomorphic q-differentials, I suspect that $H^0(C, mK)$ is the first cohomology group computed using q-differentials, but I'm not sure.

My question is that if my guess is correct, then how does one even define that cohomology (do we still have $d^m d z^m = 0$ so we have our exact sequence to define the cohomology?). And if I'm totally wrong then please tell me what that notation means.

In any case I would appreciate if someone guides me to a reference so I can read more on this object.

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It's almost certainly what Ruadhai suggests below. You can think of $H^0(C,mK)$ as sloppy way of writing $H^0(C,\mathcal{O}(mK))=H^0(C,\mathcal{O}(K)^{\otimes m})$. If you prefer to think in terms of divisors, then this is isomorphic to the space of meromorphic functions $f$ with $(f)+mK\ge 0$, where $K$ is a specified canonical divisor. – Donu Arapura Feb 11 '13 at 16:04
So does it mean that the cochains cosist of these objects Is the cohomology the usual cohomology or is a cech cohomology? I'll look into his Magama code to see if this definition makes sense. – Syed Feb 13 '13 at 16:31
Looking at Eischler Trace Formula in Breuer's thesis, I think you are certainly right. – Syed Feb 27 '13 at 15:47
up vote 3 down vote accepted

I can't access the draft you've linked but (almost always) this means the vector space of global sections of the $m^{th}$ power of the canonical bundle. Note for a Riemann surface the canonical bundle is just the cotangent bundle, the dual of the tangent bundle. For the general theory of algebraic curves, see e.g. Geometry of Algebraic Curves I by Arabello et al.

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