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

→ Main article : 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 . ${\displaystyle X}$ ${\displaystyle D\colon X\rightarrow \mathbb {Z} }$

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 : ${\displaystyle f:X\rightarrow \mathbb {P} ^{1}}$ ${\displaystyle (f)}$ ${\displaystyle x\in X}$ ${\displaystyle f}$ ${\displaystyle x}$

${\displaystyle \left(f\right)(x):={\begin{cases}0,&f{\mbox{ holomorph und ungleich null in }}x\\k,&f{\mbox{ hat eine Nullstelle von Ordnung }}k{\mbox{ in }}x\\-k,&f{\mbox{ hat eine Polstelle von Ordnung }}k{\mbox{ in }}x\\\infty ,&f{\mbox{ verschwindet in einer Umgebung von }}x\end{cases}}}$

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 . ${\displaystyle X}$ ${\displaystyle \omega }$ ${\displaystyle X}$ ${\displaystyle (\omega )}$ ${\displaystyle D}$ ${\displaystyle (\omega )}$ ${\displaystyle D=(\omega )}$

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. ${\displaystyle D}$ ${\displaystyle \textstyle \deg D:=\sum _{x\in X}D(x)}$

Statement about Riemann surfaces

Let be a compact Riemann surface of topological gender and a divisor . Then: ${\displaystyle X}$ ${\displaystyle g\in \mathbb {N} _{0}}$ ${\displaystyle D}$ ${\displaystyle X}$

${\displaystyle \ell (D)-\ell (K-D)=\deg D+1-g}$

${\displaystyle K}$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: ${\displaystyle X}$ ${\displaystyle \ell (E)}$ ${\displaystyle E}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle L(E)}$ ${\displaystyle X}$

${\displaystyle L(E):=\left\{f:X\rightarrow \mathbb {P} ^{1}{\mbox{ meromorph}}\,|\,\left(f\right)(x)\geq -E(x)\;\forall \,x\in X\right\}}$

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. ${\displaystyle X}$ ${\displaystyle K}$

It then reads:

${\displaystyle \dim _{K}H^{0}\left(X,{\mathcal {O}}_{X}\right)-\dim _{K}H^{1}\left(X,{\mathcal {O}}_{X}\right)=1-g}$

${\displaystyle {\mathcal {O}}_{X}}$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. ${\displaystyle X}$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle K=\mathbb {C} }$

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 . ${\displaystyle 0}$ ${\displaystyle \mathbb {P} ^{1}}$ ${\displaystyle {S}^{2}}$ ${\displaystyle 0}$ ${\displaystyle \mathbb {P} ^{1}}$
• 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 . ${\displaystyle \mathbb {P} ^{3}}$

Further generalizations

• Riemann-Roch-Hirzebruch theorem
• Riemann-Roch-Grothendieck theorem
• Atiyah-Singer index rate

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