TheRiemann–Roch theorem is an important theorem inmathematics, specifically incomplex analysis andalgebraic geometry, for the computation of the dimension of the space ofmeromorphic functions with prescribed zeros and allowedpoles. It relates the complex analysis of a connectedcompactRiemann surface with the surface's purely topologicalgenusg, in a way that can be carried over into purely algebraic settings.

Initially proved asRiemann's inequality byRiemann (1857), the theorem reached its definitive form for Riemann surfaces after work ofRiemann's short-lived studentGustav Roch (1865). It was later generalized toalgebraic curves, to higher-dimensionalvarieties and beyond.

ARiemann surface${\displaystyle X}$ is atopological space that is locally homeomorphic to an open subset of${\displaystyle \mathbb{C}}$, the set ofcomplex numbers. In addition, thetransition maps between these open subsets are required to beholomorphic. The latter condition allows one to transfer the notions and methods ofcomplex analysis dealing with holomorphic andmeromorphic functions on${\displaystyle \mathbb{C}}$ to the surface${\displaystyle X}$. For the purposes of the Riemann–Roch theorem, the surface${\displaystyle X}$ is always assumed to becompact. Colloquially speaking, thegenus${\displaystyle g}$ of a Riemann surface is its number ofhandles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the firstBetti number, i.e., half of the${\displaystyle \mathbb{C}}$-dimension of the firstsingular homology group${\displaystyle H_1(X,\mathbb{C})}$ with complex coefficients. The genusclassifies compact Riemann surfacesup tohomeomorphism, i.e., two such surfaces are homeomorphicif and only if their genus is the same. Therefore, the genus is an important topological invariant of a Riemann surface. On the other hand,Hodge theory shows that the genus coincides with the${\displaystyle \mathbb{C}}$-dimension of the space of holomorphic one-forms on${\displaystyle X}$, so the genus also encodes complex-analytic information about the Riemann surface.^[1]

Adivisor${\displaystyle D}$ is an element of thefree abelian group on the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients.

Any meromorphic function${\displaystyle f}$ gives rise to a divisor denoted${\displaystyle (f)}$ defined as

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

where${\displaystyle R(f)}$ is the set of all zeroes and poles of${\displaystyle f}$, and${\displaystyle s_{\nu }}$ is given by

.

The set${\displaystyle R(f)}$ is known to be finite; this is a consequence of${\displaystyle X}$ being compact and the fact that the zeros of a (non-zero) holomorphic function do not have anaccumulation point. Therefore,${\displaystyle (f)}$ is well-defined. Any divisor of this form is called aprincipal divisor. Two divisors that differ by a principal divisor are calledlinearly equivalent. The divisor of a meromorphic1-form is defined similarly. A divisor of a global meromorphic 1-form is called thecanonical divisor (usually denoted${\displaystyle K}$). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor).

The symbol${\displaystyle \mathrm{deg}(D)}$ denotes thedegree (occasionally also called index) of the divisor${\displaystyle D}$, i.e. the sum of the coefficients occurring in${\displaystyle D}$. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linearequivalence class.

The number${\displaystyle \ell (D)}$ is the quantity that is of primary interest: thedimension (over${\displaystyle \mathbb{C}}$) of the vector space of meromorphic functions${\displaystyle h}$ on the surface, such that all the coefficients of${\displaystyle (h)+D}$ are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in${\displaystyle D}$; if the coefficient in${\displaystyle D}$ at${\displaystyle z}$ is negative, then we require that${\displaystyle h}$ has a zero of at least thatmultiplicity at${\displaystyle z}$ – if the coefficient in${\displaystyle D}$ is positive,${\displaystyle h}$ can have a pole of at most that order. 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).

The Riemann–Roch theorem for a compact Riemann surface of genus${\displaystyle g}$ with canonical divisor${\displaystyle K}$ states

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

Typically, the number${\displaystyle \ell (D)}$ is the one of interest, while${\displaystyle \ell (K-D)}$ is thought of as a correction term (also called index of speciality^[2][3]) so the theorem may be roughly paraphrased by saying

dimension −correction =degree −genus + 1.

Because it is the dimension of a vector space, the correction term${\displaystyle \ell (K-D)}$ is always non-negative, so that

${\displaystyle \ell (D)\ge \mathrm{deg}(D)-g+1}$.

This is calledRiemann's inequality.Roch's part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus${\displaystyle g}$,${\displaystyle K}$ has degree${\displaystyle 2g-2}$, independently of the meromorphic form chosen to represent the divisor. This follows from putting${\displaystyle D=K}$ in the theorem. In particular, as long as${\displaystyle D}$ has degree at least${\displaystyle 2g-1}$, the correction term is 0, so that

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

The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem usingline bundles and a generalization of the theorem toalgebraic curves.

The theorem will be illustrated by picking a point${\displaystyle P}$ on the surface in question and regarding the sequence of numbers

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

i.e., the dimension of the space of functions that are holomorphic everywhere except at${\displaystyle P}$ where the function is allowed to have a pole of order at most${\displaystyle n}$. For${\displaystyle n=0}$, the functions are thus required to beentire, i.e., holomorphic on the whole surface${\displaystyle X}$. ByLiouville's theorem, such a function is necessarily constant. Therefore,${\displaystyle \ell (0)=1}$. In general, the sequence${\displaystyle \ell (n\cdot P)}$ is an increasing sequence.

TheRiemann sphere (also calledcomplex projective line) issimply connected and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of${\displaystyle \mathbb{C}}$, withtransition map being given by

${\displaystyle \mathbb{C}\setminus \{0\}\ni z\mapsto \frac{1}{z}\in \mathbb{C}\setminus \{0\}}$.

Therefore, the form${\displaystyle \omega =dz}$ on one copy of${\displaystyle \mathbb{C}}$ extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since

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

Thus, its canonical divisor is${\displaystyle K:=\mathrm{div}(\omega )=-2P}$ (where${\displaystyle P}$ is the point at infinity).

Therefore, the theorem says that the sequence${\displaystyle \ell (n\cdot P)}$ reads

1, 2, 3, ... .

This sequence can also be read off from the theory ofpartial fractions. Conversely if this sequence starts this way, then${\displaystyle g}$ must be zero.

The next case is a Riemann surface of genus${\displaystyle g=1}$, such as atorus${\displaystyle \mathbb{C}/\mathrm{\Lambda }}$, where${\displaystyle \mathrm{\Lambda }}$ is a two-dimensionallattice (a group isomorphic to${\displaystyle {\mathbb{Z}}^2}$). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate${\displaystyle z}$ on${\displaystyle C}$ yields a one-form${\displaystyle \omega =dz}$ on${\displaystyle X}$ that is everywhere holomorphic, i.e., has no poles at all. Therefore,${\displaystyle K}$, the divisor of${\displaystyle \omega }$ is zero.

On this surface, this sequence is

1, 1, 2, 3, 4, 5 ... ;

and this characterises the case${\displaystyle g=1}$. Indeed, for${\displaystyle D=0}$,${\displaystyle \ell (K-D)=\ell (0)=1}$, as was mentioned above. For${\displaystyle D=n\cdot P}$ with${\displaystyle n>0}$, the degree of${\displaystyle K-D}$ is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory ofelliptic functions.

For${\displaystyle g=2}$, the sequence mentioned above is

1, 1, ?, 2, 3, ... .

It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is ahyperelliptic curve. For${\displaystyle g>2}$ it is always true that at most points the sequence starts with${\displaystyle g+1}$ ones and there are finitely many points with other sequences (seeWeierstrass points).

Using the close correspondence between divisors andholomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: letL be a holomorphic line bundle onX. Let${\displaystyle H^0(X,L)}$ denote the space of holomorphic sections ofL. This space will be finite-dimensional; its dimension is denoted${\displaystyle h^0(X,L)}$. LetK denote thecanonical bundle onX. Then, the Riemann–Roch theorem states that

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

The theorem of the previous section is the special case of whenL is apoint bundle.

The theorem can be applied to show that there areg linearly independent holomorphic sections ofK, orone-forms onX, as follows. TakingL to be the trivial bundle,${\displaystyle h^0(X,L)=1}$ since the only holomorphic functions onX are constants. The degree ofL is zero, and${\displaystyle L^{-1}}$ is the trivial bundle. Thus,

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

Therefore,${\displaystyle h^0(X,K)=g}$, proving that there areg holomorphic one-forms.

Since the canonical bundle${\displaystyle K}$ has${\displaystyle h^0(X,K)=g}$, applying Riemann–Roch to${\displaystyle L=K}$ gives

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

which can be rewritten as

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

hence the degree of the canonical bundle is${\displaystyle \mathrm{deg}(K)=2g-2}$.

Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue inalgebraic geometry. The analogue of a Riemann surface is anon-singularalgebraic curveC over a fieldk. The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a realmanifold is two, but one as acomplex manifold. The compactness of a Riemann surface is paralleled by the condition that the algebraic curve becomplete, which is equivalent to beingprojective. Over a general fieldk, there is no good notion of singular (co)homology. The so-calledgeometric genus is defined as

${\displaystyle g(C):={\dim }_k\mathrm{\Gamma }(C,{\mathrm{\Omega }}_C^1)}$

i.e., as the dimension of the space of globally defined (algebraic) one-forms (seeKähler differential). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced byrational functions which are locally fractions ofregular functions. Thus, writing${\displaystyle \ell (D)}$ for the dimension (overk) of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient inD, the very same formula as above holds:

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

whereC is a projective non-singular algebraic curve over analgebraically closed fieldk. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into accountmultiplicities coming from the possible extensions of the base field and theresidue fields of the points supporting the divisor.^[4] Finally, for a proper curve over anArtinian 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${\displaystyle \mathcal{O}}$.^[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 areGorenstein rings, the same statement as above holds, provided that the geometric genus as defined above is replaced by thearithmetic genusg_a, defined as

${\displaystyle g_a:={\dim }_k{H}^1(C,{\mathcal{O}}_C)}$.^[6]

(For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties).^[7]

One of the important consequences of Riemann–Roch is it gives a formula for computing theHilbert polynomial of line bundles on a curve. If a line bundle${\displaystyle \mathcal{L}}$ is ample, then the Hilbert polynomial will give the first degree${\displaystyle {\mathcal{L}}^{\otimes n}}$ giving an embedding into projective space. For example, the canonical sheaf${\displaystyle {\omega }_C}$ has degree${\displaystyle 2g-2}$, which gives an ample line bundle for genus${\displaystyle g\ge 2}$.^[8] If we set${\displaystyle {\omega }_C(n)={\omega }_C^{\otimes n}}$ then the Riemann–Roch formula reads

Giving the degree${\displaystyle 1}$ Hilbert polynomial of${\displaystyle {\omega }_C}$

${\displaystyle H_{{\omega }_C}(t)=2(g-1)t-g+1}$.

Because the tri-canonical sheaf${\displaystyle {\omega }_C^{\otimes 3}}$ is used to embed the curve, the Hilbert polynomial

${\displaystyle H_C(t)=H_{{\omega }_C^{\otimes 3}}(t)}$

is generally considered while constructing theHilbert scheme of curves (and themoduli space of algebraic curves). This polynomial is

and is called theHilbert polynomial of a genus g curve.

Analyzing this equation further, the Euler characteristic reads as

Since${\displaystyle \mathrm{deg}({\omega }_C^{\otimes n})=n(2g-2)}$

${\displaystyle h^0\left(C,{\left({\omega }_C^{\otimes (n-1)}\right)}^{\vee }\right)=0}$.

for${\displaystyle n\ge 3}$, since its degree is negative for all${\displaystyle g\ge 2}$, implying it has no global sections, there is an embedding into some projective space from the global sections of${\displaystyle {\omega }_C^{\otimes n}}$. In particular,${\displaystyle {\omega }_C^{\otimes 3}}$ gives an embedding into${\displaystyle {\mathbb{P}}^N\cong \mathbb{P}(H^0(C,{\omega }_C^{\otimes 3}))}$ where${\displaystyle N=5g-5-1=5g-6}$ since${\displaystyle h^0({\omega }_C^{\otimes 3})=6g-6-g+1}$. This is useful in the construction of themoduli space of algebraic curves because it can be used as the projective space to construct theHilbert scheme with Hilbert polynomial${\displaystyle H_C(t)}$.^[9]

An irreducible plane algebraic curve of degreed has (d − 1)(d − 2)/2 − g singularities, when properly counted. It follows that, if a curve has (d − 1)(d − 2)/2 different singularities, it is arational curve and, thus, admits a rational parameterization.

TheRiemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.

Clifford's theorem on special divisors is also a consequence of the Riemann–Roch theorem. It states that for a special divisor (i.e., such that${\displaystyle \ell (K-D)>0}$) satisfying${\displaystyle \ell (D)>0}$, the following inequality holds:^[10]

${\displaystyle \ell (D)\le \frac{\mathrm{deg}D}{2}+1}$.

The statement for algebraic curves can be proved usingSerre duality. The integer${\displaystyle \ell (D)}$ is the dimension of the space of global sections of theline bundle${\displaystyle \mathcal{L}(D)}$ associated toD (cf.Cartier divisor). In terms ofsheaf cohomology, we therefore have${\displaystyle \ell (D)=\mathrm{d}\mathrm{i}\mathrm{m}{H}^0(X,\mathcal{L}(D))}$, and likewise${\displaystyle \ell ({\mathcal{K}}_X-D)=\dim H^0(X,{\omega }_X\otimes \mathcal{L}(D{)}^{\vee })}$. But Serre duality for non-singular projective varieties in the particular case of a curve states that${\displaystyle H^0(X,{\omega }_X\otimes \mathcal{L}(D{)}^{\vee })}$ is isomorphic to the dual${\displaystyle H^1(X,\mathcal{L}(D){)}^{\vee }}$. The left hand side thus equals theEuler characteristic of the divisorD. WhenD = 0, we find the Euler characteristic for the structure sheaf is${\displaystyle 1-g}$ by definition. To prove the theorem for general divisor, one can then proceed by adding points one by one to the divisor and ensure that the Euler characteristic transforms accordingly to the right hand side.

The theorem for compact Riemann surfaces can be deduced from the algebraic version usingChow's Theorem and theGAGA 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${\displaystyle \mathcal{L}(D)}$ with the sheaf${\displaystyle {\mathcal{O}}_D}$ of meromorphic functionsh such that all coefficients of the divisor${\displaystyle (h)+D}$ are nonnegative. Here the fact that the Euler characteristic transforms as desired when one adds a point to the divisor can be read off from the long exact sequence induced by the short exact sequence

${\displaystyle 0\to {\mathcal{O}}_D\to {\mathcal{O}}_{D+P}\to {\mathbb{C}}_P\to 0}$

where${\displaystyle {\mathbb{C}}_P}$ is theskyscraper sheaf atP, and the map${\displaystyle {\mathcal{O}}_{D+P}\to {\mathbb{C}}_P}$ returns the${\displaystyle -k-1}$th Laurent coefficient, where${\displaystyle k=D(P)}$.^[11]

A version of thearithmetic Riemann–Roch theorem states that ifk is aglobal field, andf is a suitably admissible function of theadeles ofk, then for everyidelea, one has aPoisson summation formula:

${\displaystyle \frac{1}{|a|}\sum _{x\in k}\hat{f}(x/a)=\sum _{x\in k}f(ax)}$.

In the special case whenk is the function field of an algebraic curve over a finite field andf is any character that is trivial onk, this recovers the geometric Riemann–Roch theorem.^[12]

Other versions of the arithmetic Riemann–Roch theorem make use ofArakelov theory to resemble the traditional Riemann–Roch theorem more exactly.

TheRiemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves byFriedrich Karl Schmidt in 1931 as he was working onperfect fields offinite characteristic. As stated byPeter 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. Actually, his proof of the Riemann–Roch theorem works for arbitrary perfect base fields, not necessarily finite.

It is foundational in the sense that the subsequent theory for curves tries to refine the information it yields (for example in theBrill–Noether theory).

There are versions in higher dimensions (for the appropriate notion ofdivisor, orline bundle). Their general formulation depends on splitting the theorem into two parts. One, which would now be calledSerre duality, interprets the${\displaystyle \ell (K-D)}$ term as a dimension of a firstsheaf cohomology group; with${\displaystyle \ell (D)}$ the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes anEuler characteristic, and the right-hand side a computation of it as adegree corrected according to the topology of the Riemann surface.

Inalgebraic geometry of dimension two such a formula was found by thegeometers of the Italian school; aRiemann–Roch theorem for surfaces was proved (there are several versions, with the first possibly being due toMax Noether).

Ann-dimensional generalisation, theHirzebruch–Riemann–Roch theorem, was found and proved byFriedrich Hirzebruch, as an application ofcharacteristic classes inalgebraic topology; he was much influenced by the work ofKunihiko Kodaira. At about the same timeJean-Pierre Serre was giving the general form of Serre duality, as we now know it.

Alexander Grothendieck proved a far-reaching generalization in 1957, now known as theGrothendieck–Riemann–Roch theorem. His work reinterprets Riemann–Roch not as a theorem about a variety, but about a morphism between two varieties. The details of the proofs were published byArmand Borel andJean-Pierre Serre in 1958.^[14] Later, Grothendieck and his collaborators simplified and generalized the proof.^[15]

Finally a general version was found inalgebraic topology, too. These developments were essentially all carried out between 1950 and 1960. After that theAtiyah–Singer index theorem opened another route to generalization. Consequently, the Euler characteristic of acoherent sheaf is reasonably computable. For just one summand within the alternating sum, further arguments such asvanishing theorems must be used.

• Arakelov theory
• Grothendieck–Riemann–Roch theorem
• Hirzebruch–Riemann–Roch theorem
• Kawasaki's Riemann–Roch formula
• Hilbert polynomial
• Moduli of algebraic curves

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• Griffiths, Phillip;Harris, Joseph (1994),Principles of algebraic geometry, Wiley Classics Library, New York:John Wiley & Sons,doi:10.1002/9781118032527,ISBN 978-0-471-05059-9,MR 1288523
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• Hartshorne, Robin (1977).Algebraic Geometry. Berlin, New York:Springer-Verlag.ISBN 978-0-387-90244-9.MR 0463157.OCLC 13348052., contains the statement for curves over an algebraically closed field. See section IV.1.
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• Vector bundles on Compact Riemann Surfaces, M. S. Narasimhan, pp. 5–6.
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• Roch, Gustav (1865)."Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen".Journal für die reine und angewandte Mathematik.1865 (64):372–376.doi:10.1515/crll.1865.64.372.S2CID 120178388.
• Schmidt, Friedrich Karl (1931),"Analytische Zahlentheorie in Körpern der Charakteristikp",Mathematische Zeitschrift,33:1–32,doi:10.1007/BF01174341,S2CID 186228993,Zbl 0001.05401, archived fromthe original on 2017-12-22, retrieved2020-05-16
• Stichtenoth, Henning (1993).Algebraic Function Fields and Codes. Springer-Verlag.ISBN 3-540-56489-6.
• Misha Kapovich,The Riemann–Roch Theorem (lecture note) an elementary introduction
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• Is there a Riemann–Roch for smooth projective curves over an arbitrary field? onMathOverflow

• Theorems in algebraic geometry
• Geometry of divisors
• Topological methods of algebraic geometry
• Theorems in complex analysis
• Bernhard Riemann

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