Editing Riemann–Roch theorem (section)

===Riemann–Roch for line bundles===
Using the close correspondence between divisors and [[holomorphic line bundle]]s on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let ''L'' be a holomorphic line bundle on ''X''. Let <math>H^0(X,L)</math> denote the space of holomorphic sections of ''L''. This space will be finite-dimensional; its dimension is denoted <math>h^0(X,L)</math>. Let ''K'' denote the [[canonical bundle]] on ''X''. Then, the Riemann–Roch theorem states that

:<math>h^0(X,L)-h^0(X,L^{-1}\otimes K)=\deg(L)+1-g</math>.

The theorem of the previous section is the special case of when ''L'' is a [[point bundle]].

The theorem can be applied to show that there are ''g'' linearly independent holomorphic sections of ''K'', or [[one-form]]s on ''X'', as follows. Taking ''L'' to be the trivial bundle, <math> h^0(X,L)=1</math> since the only holomorphic functions on ''X'' are constants. The degree of ''L'' is zero, and <math>L^{-1}</math> is the trivial bundle. Thus,

:<math>1-h^0(X,K)=1-g</math>.

Therefore, <math>h^0(X,K)=g</math>, proving that there are ''g'' holomorphic one-forms.