Riemann-Roch theorem
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In algebraic geometry the Riemann-Roch theorem states that if is a smooth algebraic curve, and is an invertible sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then the the following properties hold:
- The Euler characteristic of is given by
- There is a canonical isomorphism
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.