I am reading the draft of "[Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups](http://www.math.uga.edu/~davids/ivrg/SwinarskiEquations.pdf)" 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.



Reference for notation $H^0(C, mK)$
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.


