Riemann-Roch theorem

In algebraic geometry the Riemann-Roch theorem states that if $C$ is a smooth algebraic curve, and ${\mathcal {L}}$ is an invertible sheaf on $C$ then the the following properties hold:

• The Euler characteristic of ${\mathcal {L}}$ is given by $h^{0}({\mathcal {L}})-h^{1}({\mathcal {L}})=deg({\mathcal {L}})-(g-1)$
• There is a canonical isomorphism $H^{0}(L)(K_{C}\otimes {\mathcal {L}}^{\vee })\cong H^{1}({\mathcal {L}})$

Generalizations

• Riemann-Roch for surfaces and Noether's formula
• Hirzebruch-Riemann-Roch theorem
• Grothendieck-Riemann-Roch theorem
• Atiya-Singer index theorem

Proofs

Using modern tools, the theorem is an immediate consequence of Serre's duality.