Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles.
The latter condition allows one to transfer the notions and methods of complex analysis dealing with holomorphic and meromorphic functions on
Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients.
being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an accumulation point.
Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in
The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).
Roch's part of the statement is the description of the possible difference between the sides of the inequality.
on the surface in question and regarding the sequence of numbers i.e., the dimension of the space of functions that are holomorphic everywhere except at
The Riemann sphere (also called complex projective line) is simply connected and hence its first singular homology is zero.
extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since Thus, its canonical divisor is
Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right.
The theorem can be applied to show that there are g linearly independent holomorphic sections of K, or one-forms on X, as follows.
Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in algebraic geometry.
The compactness of a Riemann surface is paralleled by the condition that the algebraic curve be complete, which is equivalent to being projective.
for the dimension (over k) of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient in D, the very same formula as above holds: where C is a projective non-singular algebraic curve over an algebraically closed field k. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account multiplicities coming from the possible extensions of the base field and the residue fields of the points supporting the divisor.
[4] Finally, for a proper curve over an Artinian ring, the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf
[5] The smoothness assumption in the theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are Gorenstein rings, the same statement as above holds, provided that the geometric genus as defined above is replaced by the arithmetic genus ga, defined as (For smooth curves, the geometric genus agrees with the arithmetic one.)
[7] One of the important consequences of Riemann–Roch is it gives a formula for computing the Hilbert polynomial of line bundles on a curve.
[9] An irreducible plane algebraic curve of degree d has (d − 1)(d − 2)/2 − g singularities, when properly counted.
The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.
, the following inequality holds:[10] The statement for algebraic curves can be proved using Serre duality.
But Serre duality for non-singular projective varieties in the particular case of a curve states that
The theorem for compact Riemann surfaces can be deduced from the algebraic version using Chow's Theorem and the GAGA principle: in fact, every compact Riemann surface is defined by algebraic equations in some complex projective space.
(Chow's Theorem says that any closed analytic subvariety of projective space is defined by algebraic equations, and the GAGA principle says that sheaf cohomology of an algebraic variety is the same as the sheaf cohomology of the analytic variety defined by the same equations).
One may avoid the use of Chow's theorem by arguing identically to the proof in the case of algebraic curves, but replacing
As stated by Peter Roquette,[13] The first main achievement of F. K. Schmidt is the discovery that the classical theorem of Riemann–Roch on compact Riemann surfaces can be transferred to function fields with finite base field.
There are versions in higher dimensions (for the appropriate notion of divisor, or line bundle).
the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes an Euler characteristic, and the right-hand side a computation of it as a degree corrected according to the topology of the Riemann surface.
In algebraic geometry of dimension two such a formula was found by the geometers of the Italian school; a Riemann–Roch theorem for surfaces was proved (there are several versions, with the first possibly being due to Max Noether).
An n-dimensional generalisation, the Hirzebruch–Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic classes in algebraic topology; he was much influenced by the work of Kunihiko Kodaira.
