Riemann-Roch Theorem

The Riemann-Roch theorem is an important theorem of mathematics , especially in complex analysis and algebraic geometry , which helps in calculating the dimension of the space of meromorphic functions with prescribed zeros and allowed poles . The theorem connects a complex analysis of connected compact Riemann surfaces with a purely topological genus of the surface g , using methods that can be extended to purely algebraic situations.

Originally proved by Riemann as the Riemann inequality ^[1] , the theorem got its final form for Riemann surfaces after the work of the early deceased student Riemann Gustav Roch ^[2] . Later, the theorem was generalized to algebraic curves , to manifolds of higher dimension, and so on.

Content

1 Preliminary remarks
2 statement of the theorem
- 2.1 Examples
  - 2.1.1 Genus 0
  - 2.1.2 Genus 1
  - 2.1.3 Genus 2 and above
- 2.2 Riemann-Roch theorem for line bundles
- 2.3 Riemann-Roch Theorem for Algebraic Curves
3 Proof
4 Applications
5 Generalizations of the Riemann - Roch Theorem
6 notes
7 Literature

Preliminaries

Riemann surface of genus 3.

A Riemann surface X is a topological space that is locally homeomorphic to an open subset of C , the set of complex numbers. In addition, the transition functions between these open subsets are required to be holomorphic . The last condition allows you to transfer the terms and methods of complex analysis dealing with holomorphic and meromorphic functions on C to the surface X. For the purposes of the Riemann – Roch theorem, the surface X is always assumed to be compact . Roughly speaking, the genus g of a Riemann surface is the number of surface handles. For example, the genus of the Riemann surface shown on the right is three. More precisely, the genus is defined as half the first Betty number , that is, half the complex dimension of the first group of singular homologies H ₁ ( X , C ) with complex coefficients. A genus classifies compact Riemann surfaces up to homeomorphism , that is, two such surfaces are homeomorphic if and only if their genus coincides. Thus, the genus is an important topological invariant of a Riemann surface. On the other hand, Hodge's theory shows that the genus coincides with the (complex) dimension of the space of holomorphic 1-forms on X , so that the genus also encodes complex-analytic information about the Riemann surface ^[3] .

The divisor D is an element of a free abelian group generated by points on a surface. Equivalently, a divisor is a finite linear combination with integer coefficients of surface points.

Any meromorphic function f gives a divisor denoted by ( f ), which is defined as

(f):=\sum _{z_{\nu }\in R(f)}s_{\nu }z_{\nu }

{\ displaystyle (f): = \ sum _ {z _ {\ nu} \ in R (f)} s _ {\ nu} z _ {\ nu}}

(f):=\sum _{z_{\nu }\in R(f)}s_{\nu }z_{\nu }

where R ( f ) is the set of all zeros and poles of the function f , and s _ν is defined as follows

s_{\nu }:=a

{\ displaystyle s _ {\ nu}: = a}

, if

z_{\nu }

{\ displaystyle z _ {\ nu}}

is a zero of order a , and -a if

z_{\nu }

{\ displaystyle z _ {\ nu}}

is a pole of order a.

It is known that the set R ( f ) is finite. This is a consequence of the compactness of X and the fact that the zeros of a (nonzero) holomorphic function have no limit points . Thus, ( f ) is well defined. Any divisor of this kind is called the main divisor. Two divisors that differ by the main divisor are called linearly equivalent . The divisor of a meromorphic 1-form is defined similarly. The divisor of a global meromorphic 1-form is called the (usually denoted by K ). Any two meromorphic 1-forms give linearly equivalent divisors, so that the canonical divisor is uniquely defined up to linear equivalence.

The symbol deg ( D ) means the degree (sometimes called the index) of the divisor D , that is, the sum of the coefficients encountered in D. It can be shown that the divisor of a global meromorphic function always has degree 0, so that the degree of the divisor depends only on the linear equivalence class.

Number $\ell (D)$ ${\ displaystyle \ ell (D)}$ is the quantity of main interest - the dimension (over C ) of the vector space of meromorphic functions h on the surface such that all the coefficients of the divisor ( h ) + D are non-negative. Intuitively, we can think of them as meromorphic functions whose poles at each point are no worse than the corresponding coefficients D. If the coefficient at D at z is negative, then we require that h have degree zero of at least at z , if the coefficient at D is positive, h can have a pole not exceeding this order. Vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication by a global meromorphic function (which is uniquely defined up to a scalar).

Statement of the theorem

The Riemann – Roch theorem for a compact Riemann surface of genus g with the canonical divisor K states that

\ell (D)-\ell (K-D)=\deg(D)-g+1.

{\ displaystyle \ ell (D) - \ ell (KD) = \ deg (D) -g + 1.}

Usually a number $\ell (D)$ ${\ displaystyle \ ell (D)}$ is the desired number while $\ell (K-D)$ ${\ displaystyle \ ell (KD)}$ considered as a correcting term (also called the specialty index ^[4] ^[5] ), so the theorem can be roughly rephrased by saying

dimension - correction = degree - gender + 1.

Correction member $\ell (K-D)$ ${\ displaystyle \ ell (KD)}$ always non-negative so that

\ell (D)\geq \deg(D)-g+1.

{\ displaystyle \ ell (D) \ geq \ deg (D) -g + 1.}

This expression is called the Riemann inequality . Roch's contribution to this statement is to describe the possible difference between the two parts of the inequality. On a Riemann surface of general form of genus g , K has degree 2 g - 2. This can be obtained by setting D = K. In the theorem, in particular, if D has degree at least 2 g - 1, the correcting term is 0, so

\ell (D)=\deg(D)-g+1.

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

There are also a number of other closely related theorems - the equivalent statement of the theorem using and the generalization of the theorem to algebraic curves .

Examples

The theorem can be illustrated by choosing the point P on the surface under consideration and considering the sequence of numbers

\ell (n\cdot P),n\geq 0

{\ displaystyle \ ell (n \ cdot P), n \ geq 0}

that is, the dimensions of the space of functions holomorphic everywhere except for the point P , at which the functions are allowed to have a pole of order not exceeding n . For n = 0, the functions must then be integer , i.e. holomorphic on the whole surface X. By Liouville's theorem, such a function must be constant. In this way, $\ell (0)=1$ ${\ displaystyle \ ell (0) = 1}$ . In general, the sequence $\ell (n\cdot P)$ ${\ displaystyle \ ell (n \ cdot P)}$ increasing.

Genus 0

The Riemann sphere (also called the complex projective line) is simply connected , and therefore its first singular homology is equal to zero. In particular, her gender is zero. The sphere can be covered with two copies of C with the transition function defined by the expression

\mathbf {C} ^{\times }\ni z\mapsto 1/z.

{\ displaystyle \ mathbf {C} ^ {\ times} \ ni z \ mapsto 1 / z.}

Thus, the form ω = d z on one copy of C extends to a meromorphic form on the Riemann sphere - it has a double pole at infinity, since

d\left({\frac {1}{z}}\right)=-{\frac {1}{z^{2}}}\,dz.

{\ displaystyle d \ left ({\ frac {1} {z}} \ right) = - {\ frac {1} {z ^ {2}}} \, dz.}

Then its divisor K : = div ( ω ) = −2 P (where P is a point at infinity).

Thus, the theorem states that the sequence $\ell (n\cdot P)$ ${\ displaystyle \ ell (n \ cdot P)}$ has the form

1, 2, 3, ....

The same sequence can be deduced from the theory of decomposition into elementary fractions . Conversely, if a sequence starts in this way, g must be zero.

Genus 1

Thor.

The next case is Riemann surfaces of genus g = 1, such as the torus C / Λ, where Λ is a two-dimensional lattice (a group isomorphic to Z ² ). Its genus is equal to unity — its first group of singular homologies is freely generated by two loops, as shown in the figure to the right. The standard complex coordinate z on C gives a 1-form ω = d z on X , which is holomorphic everywhere, that is, has no poles at all. Therefore, K , the divisor ω, is equal to zero.

On this surface, the sequence will be

1, 1, 2, 3, 4, 5 ...;

and this characterizes the case g = 1. Moreover, for $D=0,\ell (K-D)=\ell (0)=1$ ${\ displaystyle D = 0, \ ell (KD) = \ ell (0) = 1}$ as mentioned above. For D = nP with n > 0, the degree of K - D is strictly negative, so that the correction term is zero. A sequence of dimensions can also be deduced from the theory of elliptic functions .

Genus 2 and above

For g = 2, the sequence mentioned above will be

1, 1,?, 2, 3, ....

Is there a member? degree 2 is 1 or 2 depending on the point. It can be proved that on any curve of genus 2 there are exactly six points with a sequence of 1, 1, 2, 2, ..., and the remaining points have a sequence of 1, 1, 1, 2, ... In particular, a curve of genus 2 is a . For g > 2, it is always true that the sequence of most points starts with g + 1 units and there is a finite number of points with other sequences (see ).

Riemann-Roch theorem for line bundles

Using the close correspondence between divisors and on a Riemann surface, we can state the theorem in a different, but still equivalent form. Let L be a holomorphic line bundle on X. Let be $H^{0}(X,L)$ ${\ displaystyle H ^ {0} (X, L)}$ denotes the space of holomorphic sections L. This space will be finite-dimensional and this dimension is denoted as $h^{0}(X,L)$ ${\ displaystyle h ^ {0} (X, L)}$ . Let K denote the on X. Then the Riemann-Roch theorem states that

h^{0}(X,L)-h^{0}(X,L^{-1}\otimes K)=\deg(L)+1-g.

{\ displaystyle h ^ {0} (X, L) -h ^ {0} (X, L ^ {- 1} \ otimes K) = \ deg (L) + 1-g.}

The theorem from the previous section is a special case when L is a point bundle.

The theorem can be used to show that there exist g holomorphic sections of K or 1-forms on X. If we take the trivial bundle as L , we obtain $h^{0}(X,L)=1$ ${\ displaystyle h ^ {0} (X, L) = 1}$ , since only constant functions on X are holomorphic. The degree of L is zero and $L^{-1}$ ${\ displaystyle L ^ {- 1}}$ is a trivial bundle. Then

1-h^{0}(X,K)=1-g.

{\ displaystyle 1-h ^ {0} (X, K) = 1-g.}

In this way, $h^{0}(X,K)=g$ ${\ displaystyle h ^ {0} (X, K) = g}$ , which proves that there are g holomorphic 1-forms.

Riemann-Roch Theorem for Algebraic Curves

Each term in the above statement of the Riemann – Roch theorem for divisors on Riemann surfaces has an analogue in algebraic geometry . An analogue of the Riemann surface is a nonsingular algebraic curve C over the field k . A difference in terminology (curves instead of surfaces) arises because the dimension of a Riemann surface as a real manifold is equal to two, and as a complex manifold to unity. The compactness of the Riemann surface is due to the condition that the algebraic curve , which is equivalent to its . Over a field k of general form, there is no good concept of singular (co) homology. The so-called geometric genus is defined as

g(C):=\dim _{k}\Gamma (C,\Omega _{C}^{1})

{\ displaystyle g (C): = \ dim _ {k} \ Gamma (C, \ Omega _ {C} ^ {1})}

that is, as the dimension of the space of globally defined (algebraic) 1-forms (see Kähler differential ). Finally, meromorphic functions on a Riemann surface are locally represented as quotients of holomorphic functions. Consequently, they are replaced by rational functions that are locally partial of . Thus, if we denote by $\ell (D)$ ${\ displaystyle \ ell (D)}$ the dimension (over k ) of the space of rational functions on a curve whose poles at each point are no worse than the corresponding coefficients in D , the same formula holds as above:

\ell (D)-\ell (K-D)=\deg(D)-g+1.

{\ displaystyle \ ell (D) - \ ell (KD) = \ deg (D) -g + 1.}

where C is a projective nonsingular algebraic curve over an algebraically closed field k . In fact, the same formula holds for projective curves over any field, except that when calculating the degree of the divisor it is necessary to take into account the points ^[6] . Finally, for a suitable curve over an artinian ring , the Euler characteristic of the line bundle associated with the divisor is given by the degree of the divisor (properly defined), plus the Euler characteristic of the structural bundle ${\mathcal {O}}$ ${\ displaystyle {\ mathcal {O}}}$ ^[7] .

The smoothness assumption in the theorem can also be weakened - for the (projective) curve over an algebraically closed field, all local rings of which are , the same statement holds as above, except that the geometric genus is replaced by the g _a defined as

g_{a}:=\dim _{k}H^{1}(C,{\mathcal {O}}_{C}).

{\ displaystyle g_ {a}: = \ dim _ {k} H ^ {1} (C, {\ mathcal {O}} _ {C}).}

^[8]

(For smooth curves, the geometrical genus coincides with the arithmetic one.) The theorem was also extended to general singular curves (and manifolds of higher dimension) ^[9] .

Proof

The statement for algebraic curves can be proved using Serre duality . The integer I ( D ) is the dimension of the space of global sections of the ${\mathcal {L}}(D)$ ${\ displaystyle {\ mathcal {L}} (D)}$ associated with D. In terms of the cohomology of sheaves , we therefore have $I(D)=\mathrm {dim} H^{0}(X,{\mathcal {L}}(D))$ ${\ displaystyle I (D) = \ mathrm {dim} H ^ {0} (X, {\ mathcal {L}} (D))}$ and in the same way $I({\mathcal {K}}_{X}-D)=\dim H^{0}(X,\omega _{X}\otimes {\mathcal {L}}(D)^{\vee })$ ${\ displaystyle I ({\ mathcal {K}} _ {X} -D) = \ dim H ^ {0} (X, \ omega _ {X} \ otimes {\ mathcal {L}} (D) ^ { \ vee})}$ . However, Serre duality for nonsingular projective varieties in the particular case of a curve states that $H^{0}(X,\omega _{X}\otimes {\mathcal {L}}(D)^{\vee })$ ${\ displaystyle H ^ {0} (X, \ omega _ {X} \ otimes {\ mathcal {L}} (D) ^ {\ vee})}$ isomorphic to dual space $\simeq H^{1}(X,{\mathcal {L}}(D))^{\vee }$ ${\ displaystyle \ simeq H ^ {1} (X, {\ mathcal {L}} (D)) ^ {\ vee}}$ . The left-hand side is then equal to the Euler characteristic of the divisor D. If D = 0, we find the Euler characteristic of the structural beam, which is equal to $1-g$ ${\ displaystyle 1-g}$ a-priory. To prove the theorem for general divisors, we can add points one after another to the divisor and delete some and prove that the Euler characteristic is transformed according to the right side.

The theorem for compact Riemann surfaces can be deduced from the algebraic version using and the principle (Géometrie Algébrique et Géométrie Analytique). In fact, any compact Riemann surface is defined by algebraic equations in some complex projective space. (Zhou's theorem states that any closed analytic submanifold of a projective space is defined by algebraic equations, and the GAGA principle states that the cohomology of sheaves of an algebraic variety is the same as the cohomology of sheaves of an analytic variety defined by some equations).

Applications

An irreducible plane algebraic curve of degree d has $(d-1)(d-2)/2-g$ ${\ displaystyle (d-1) (d-2) / 2-g}$ singular points, if considered appropriately. It follows that if the curve has $(d-1)(d-2)/2$ ${\ displaystyle (d-1) (d-2) / 2}$ different singular points, this is a rational curve and admitting rational parameterization.

, относяющаяся к (разветвлённым) отображениям между римановыми поверхностями или алгебраическими кривыми, является следствием теоремы Римана — Роха.

является также следствием теоремы Римана — Роха. Она утверждает, что для специального дивизора (то есть такого, что $\ell (K-D)>0$ $\ell (KD)>0$ ), удовлетворяющего условию $\ell (D)>0$ $\ell (D)>0$ , выполняется следующее ^[10] :

\ell (D)\leq {\frac {\deg D}{2}}+1.

\ell (D)\leq {\frac {\deg D}{2}}+1.

Обобщения теоремы Римана — Роха

Теорему Римана — Роха для кривых доказали для римановых поверхностей Риман и Рох в 1850-х годах, а для алгебраических кривых доказал Фридрих Карл Шмидт в 1931 году, работая с совершенными полями конечной характеристики . Согласно Петру Рокетту :

Первое большое достижение Ф. К. Шмидта — открытие факта, что классическая теорема Римана — Роха на компактных римановых поверхностях может быть перенесена на поле функций с конечным базовым полем. Фактически, его доказательство теоремы Римана — Роха работает для произвольных совершенных базовых полей, не обязательно конечных.

Теорема является фундаментальной в том смысле, что более поздняя теория для кривых пытается усовершенствовать информацию, получаемую из теоремы (например, в теории Брилля-Нётера ).

Имеются версии для более высоких размерностей (при подходящем понятии дивизора или ). Их формулировка зависит от разбиения теоремы на две части. Первая, теперь называемая , интерпретирует член $\ell (K-D)$ $\ell (KD)$ как размерность первой группы когомологий пучков . At $\ell (D)$ $\ell (D)$ , равном размерности нулевой группы когомологий или пространства сечений, левая часть теоремы становится эйлеровой характеристикой , а правая часть становится формулой вычисления её как степени , исправленной согласно топологии римановой поверхности.

В алгебраической геометрии размерности два такая формула была найдена геометрами итальянской школы . Теорема Римана — Роха для поверхностей была доказана (существует несколько версий, первое доказательство принадлежит Максу Нётеру ). Такое положение вещей сохранялось примерно до 1950 года.

Обобщение для n -мерных многообразий, , было доказано Фридрихом Хирцебрухом как приложение характеристических классов из алгебраической топологии . На Хирцебруха повлияла работа Кунихико Кодайра . Примерно в то же время Жан-Пьер Серр дал общую форму , как мы её теперь знаем.

Александр Гротендик доказал далеко идущее обобщение в 1957 году, известное сейчас как . Его работа даёт другое толкование теоремы Римана — Роха, не как теоремы о многообразии, а как теоремы о морфизме между двумя многообразиями. Детали доказательства опубликовали Борель и Серр в 1958 году.

Наконец, общая версия была также найдена в алгебраической топологии . Эти исследования, в основном, проведены между 1950 и 1960 годами. После этого теорема Атьи — Зингера об индексе открыла другие пути обобщения.

Результатом является факт, что эйлерова характеристика ( когерентного пучка ) иногда вполне вычислима. Если требуется вычислить отдельный член суммы, должны быть использованы другие аргументы, такие как теоремы об обращении в нуль.

Notes

↑ Riemann, 1857 .
↑ Roch, 1865 .
↑ Griffiths, Harris, 1994 , с. 116, 117.
↑ Stichtenoth, 1993 , с. 22.
↑ Mukai, 2003 , с. 295–297.
↑ Liu, 2002 , с. Section 7.3.
↑ Altman, Kleiman, 1970 , с. 164, Theorem VIII.1.4..
↑ Hartshorne, 1986 , с. 375–386.
↑ Baum, Fulton, MacPherson, 1975 , с. 101–145.
↑ Fulton, 1989 , с. 109.

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[_7fc42fe60acae512-1] Riemann, 1857 .

[_3c98213f32352c1d-2] Roch, 1865 .

[_0a753a447585f3e4-3] Griffiths, Harris, 1994 , с. 116, 117.

[_c2a65c1847f7e874-4] Stichtenoth, 1993 , с. 22.

[_6c1c5e59d1c36bb0-5] Mukai, 2003 , с. 295–297.

[_c5a42fb68c29df10-6] Liu, 2002 , с. Section 7.3.

[_75742100a37d7c03-7] Altman, Kleiman, 1970 , с. 164, Theorem VIII.1.4..

[_90519c8d1db5d618-8] Hartshorne, 1986 , с. 375–386.

[_55a002344db1c92b-9] Baum, Fulton, MacPherson, 1975 , с. 101–145.

[_680aa370d3e5343a-10] Fulton, 1989 , с. 109.