Riemann-Roch theorem
The Riemann-Roch theorem (after the mathematician Bernhard Riemann and his student Gustav Roch ) is a central statement of the theory of compact Riemann surfaces . It indicates how many linearly independent meromorphic functions with given zeros and poles exist on a compact Riemann surface. The theorem was later extended to algebraic curves , generalized even further, and is also being further developed in current research.
divisor
In order to be able to prescribe zero and pole positions of a function at certain points, the term divisor is introduced. Let be a Riemann surface. A function is called a divisor if it only differs from zero at isolated points .
The divisor of a meromorphic function is denoted by and is defined in such a way that the zero or pole order of in is assigned to each point :
Thus, the divisor of a function is actually a divisor according to the first definition, if the function is different from the null function on each connected component . For a meromorphic 1-form on the divisor is defined as for a function. A divisor is called a canonical divisor if it can be written as a divisor of a meromorphic 1-form , i.e. if .
For a compact Riemann area, the degree of a divisor is defined by . The sum is finite, because due to the compactness of the carrier from isolated points it must be a finite set.
Statement about Riemann surfaces
Let be a compact Riemann surface of topological gender and a divisor . Then:
stands for any canonical divisor . denotes for a divisor the dimension of the vector space of the meromorphic functions to whose zero and pole positions are restricted by the divisor as follows:
Statement about algebraic curves
For non-singular projective algebraic curves over an algebraically closed field , the Riemann-Roch theorem is usually formulated with the help of the cohomology theory.
It then reads:
is the sheaf of regular features on . Instead of the topological gender, the arithmetic gender of the curve occurs, which in the case coincides with the topological gender . The duality theorem of Serre states that the formulation in the case with that matches the section on Riemannian surfaces.
Consequences
- As a first classification result, it immediately follows that every Riemann surface is isomorphic by gender to the Riemann sphere , so in particular only one holomorphic structure can be defined on the sphere . For non-singular projective curves of gender it applies accordingly that they are birationally equivalent .
- The Riemann-Hurwitz formula about the mapping behavior of holomorphic functions between two compact Riemann surfaces or about the mapping behavior of morphisms between two non-singular projective curves.
- An embedding theorem: Every compact Riemann surface or every non-singular projective curve can be embedded in the projective space .
Further generalizations
literature
- Otto Forster : Riemann surfaces (= Heidelberg pocket books 184). Springer, Berlin et al. 1977, ISBN 3-540-08034-1 (English: Lectures on Riemann Surfaces (= Graduate Texts in Mathematics 81). Corrected 2nd printing. Ibid 1991, ISBN 3-540-90617-7 ).
- Robin Hartshorne : Algebraic Geometry (= Graduate Texts in Mathematics 52). Springer, New York et al. 1977, ISBN 0-387-90244-9 .
- Klaus Lamotke : Riemann surfaces. Springer, Berlin et al. 2005, ISBN 3-540-57053-5 .
- Jeremy Gray, The Riemann-Roch theorem and geometry, 1854-1914 , ICM Berlin 1998, Documenta Mathematica, 1998, pp. 811-822