The Riemann–Roch theorem is an important theorem in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, for the computation of the dimension of the space of
meromorphic functions with prescribed zeros and allowed
poles
Pole or poles may refer to:
People
*Poles (people), another term for Polish people, from the country of Poland
* Pole (surname), including a list of people with the name
* Pole (musician) (Stefan Betke, born 1967), German electronic music artist
...
. It relates the complex analysis of a connected
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Riemann surface with the surface's purely topological
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'', in a way that can be carried over into purely algebraic settings.
Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of
Riemann's short-lived student . It was later generalized to
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s, to higher-dimensional
varieties and beyond.
Preliminary notions

A
Riemann surface is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is locally homeomorphic to an open subset of
, the set of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. In addition, the
transition map
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies t ...
s between these open subsets are required to be
holomorphic. The latter condition allows one to transfer the notions and methods of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
dealing with holomorphic and
meromorphic functions on
to the surface
. For the purposes of the Riemann–Roch theorem, the surface
is always assumed to be
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
. Colloquially speaking, the
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
of a Riemann surface is its number of
handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first
Betti number, i.e., half of the
-dimension of the first
singular homology group
with complex coefficients. The genus
classifies compact Riemann surfaces
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
, i.e., two such surfaces are homeomorphic
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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
-dimension of the space of holomorphic one-forms on
, so the genus also encodes complex-analytic information about the Riemann surface.
A
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
is an element of the
free 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
gives rise to a divisor denoted
defined as
:
where
is the set of all zeroes and poles of
, and
is given by
:
.
The set
is known to be finite; this is a consequence of
being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an
accumulation point. Therefore,
is well-defined. Any divisor of this form is called a
principal divisor. Two divisors that differ by a principal divisor are called
linearly equivalent. The divisor of a meromorphic
1-form
In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the t ...
is defined similarly. A divisor of a global meromorphic 1-form is called the
canonical divisor
The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical examp ...
(usually denoted
). 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
denotes the ''degree'' (occasionally also called index) of the divisor
, i.e. the sum of the coefficients occurring in
. 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 linear
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
.
The number
is the quantity that is of primary interest: the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
(over
) of the vector space of meromorphic functions
on the surface, such that all the coefficients of
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
; if the coefficient in
at
is negative, then we require that
has a zero of at least that
multiplicity at
– if the coefficient in
is positive,
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).
Statement of the theorem
The Riemann–Roch theorem for a compact Riemann surface of genus
with canonical divisor
states
:
.
Typically, the number
is the one of interest, while
is thought of as a correction term (also called index of speciality) 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
is always non-negative, so that
:
.
This is called ''Riemann'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
,
has degree
, independently of the meromorphic form chosen to represent the divisor. This follows from putting
in the theorem. In particular, as long as
has degree at least
, the correction term is 0, so that
:
.
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 using
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
s and a generalization of the theorem to
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s.
Examples
The theorem will be illustrated by picking a point
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
where the function is allowed to have a pole of order at most
. For
, the functions are thus required to be
entire, i.e., holomorphic on the whole surface
. By
Liouville's theorem, such a function is necessarily constant. Therefore,
. In general, the sequence
is an increasing sequence.
Genus zero
The
Riemann sphere (also called
complex projective line) is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of
, with
transition map
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies t ...
being given by
:
.
Therefore, the form
on one copy of
extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since
:
Thus, its canonical divisor is
(where
is the point at infinity).
Therefore, the theorem says that the sequence
reads
: 1, 2, 3, ... .
This sequence can also be read off from the theory of
partial fractions. Conversely if this sequence starts this way, then
must be zero.
Genus one

The next case is a Riemann surface of genus
, such as a
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
, where
is a two-dimensional
lattice (a group isomorphic to
). 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
on
yields a one-form
on
that is everywhere holomorphic, i.e., has no poles at all. Therefore,
, the divisor of
is zero.
On this surface, this sequence is
:1, 1, 2, 3, 4, 5 ... ;
and this characterises the case
. Indeed, for
,
, as was mentioned above. For
with
, the degree of
is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of
elliptic functions.
Genus two and beyond
For
, 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 a
hyperelliptic curve. For
it is always true that at most points the sequence starts with
ones and there are finitely many points with other sequences (see
Weierstrass points).
Riemann–Roch for line bundles
Using the close correspondence between divisors and
holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let ''L'' be a holomorphic line bundle on ''X''. Let
denote the space of holomorphic sections of ''L''. This space will be finite-dimensional; its dimension is denoted
. Let ''K'' denote the
canonical bundle on ''X''. Then, the Riemann–Roch theorem states that
:
.
The theorem of the previous section is the special case of when ''L'' is a
point bundle.
The theorem can be applied to show that there are ''g'' linearly independent holomorphic sections of ''K'', or
one-forms on ''X'', as follows. Taking ''L'' to be the trivial bundle,
since the only holomorphic functions on ''X'' are constants. The degree of ''L'' is zero, and
is the trivial bundle. Thus,
:
.
Therefore,
, proving that there are ''g'' holomorphic one-forms.
Degree of canonical bundle
Since the canonical bundle
has
, applying Riemann–Roch to
gives
:
which can be rewritten as
:
hence the degree of the canonical bundle is
.
Riemann–Roch theorem for algebraic curves
Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. The analogue of a Riemann surface is a
non-singular algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
''C'' over a field ''k''. The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a real
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
is two, but one as a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
. The compactness of a Riemann surface is paralleled by the condition that the algebraic curve be
complete, which is equivalent to being
projective. Over a general field ''k'', there is no good notion of singular (co)homology. The so-called
geometric genus is defined as
:
i.e., as the dimension of the space of globally defined (algebraic) one-forms (see
Kähler differential). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced by
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s which are locally fractions of
regular function
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a reg ...
s. Thus, writing
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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
''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. Finally, for a proper curve over an
Artinian ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
, 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
.
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 ''g''
''a'', defined as
:
.
(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).
Applications
Hilbert polynomial
One of the important consequences of Riemann–Roch is it gives a formula for computing the
Hilbert polynomial of line bundles on a curve. If a line bundle
is ample, then the Hilbert polynomial will give the first degree
giving an embedding into projective space. For example, the canonical sheaf
has degree
, which gives an ample line bundle for genus
. If we set
then the Riemann–Roch formula reads
:
Giving the degree
Hilbert polynomial of
:
.
Because the tri-canonical sheaf
is used to embed the curve, the Hilbert polynomial
is generally considered while constructing the
Hilbert scheme of curves (and the
moduli space of algebraic curves). This polynomial is
and is called the Hilbert polynomial of a genus g curve.
Pluricanonical embedding
Analyzing this equation further, the Euler characteristic reads as
:
Since
:
.
for
, since its degree is negative for all
, implying it has no global sections, there is an embedding into some projective space from the global sections of
. In particular,
gives an embedding into
where
since
. This is useful in the construction of the
moduli space of algebraic curves because it can be used as the projective space to construct the
Hilbert scheme with Hilbert polynomial
.
Genus of plane curves with singularities
An irreducible plane algebraic curve of degree ''d'' 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 a
rational curve and, thus, admits a rational parameterization.
Riemann–Hurwitz formula
The
Riemann–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
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
) satisfying
, the following inequality holds:
:
.
Proof
Proof for algebraic curves
The statement for algebraic curves can be proved using
Serre duality. The integer
is the dimension of the space of global sections of the
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
associated to ''D'' (''cf.''
Cartier divisor). In terms of
sheaf cohomology, we therefore have
, and likewise
. But Serre duality for non-singular projective varieties in the particular case of a curve states that
is isomorphic to the dual
. The left hand side thus equals the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of the divisor ''D''. When ''D'' = 0, we find the Euler characteristic for the structure sheaf is
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.
Proof for compact Riemann surfaces
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
with the sheaf
of meromorphic functions ''h'' such that all coefficients of the divisor
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
:
where
is the
skyscraper sheaf at ''P'', and the map
returns the
th Laurent coefficient, where
.
Arithmetic Riemann–Roch theorem
A version of the
arithmetic Riemann–Roch theorem states that if ''k'' is a
global field, and ''f'' is a suitably admissible function of the
adeles of ''k'', then for every
idele ''a'', one has a
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the pe ...
:
:
.
In the special case when ''k'' is the function field of an algebraic curve over a finite field and ''f'' is any character that is trivial on ''k'', this recovers the geometric Riemann–Roch theorem.
Other versions of the arithmetic Riemann–Roch theorem make use of
Arakelov theory to resemble the traditional Riemann–Roch theorem more exactly.
Generalizations of the Riemann–Roch theorem
The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by
Friedrich Karl Schmidt in 1931 as he was working on
perfect fields of
finite characteristic. As stated by
Peter Roquette,
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 the
Brill–Noether theory).
There are versions in higher dimensions (for the appropriate notion of
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
, or
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
). Their general formulation depends on splitting the theorem into two parts. One, which would now be called
Serre duality, interprets the
term as a dimension of a first
sheaf cohomology group; with
the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes an
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
, and the right-hand side a computation of it as a ''degree'' corrected according to the topology of the Riemann surface.
In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the ...
).
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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
; he was much influenced by the work of
Kunihiko Kodaira. At about the same time
Jean-Pierre Serre was giving the general form of Serre duality, as we now know it.
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
proved a far-reaching generalization in 1957, now known as the
Grothendieck–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 by
Armand Borel and
Jean-Pierre Serre in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof.
[SGA 6, Springer-Verlag (1971).]
Finally a general version was found in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, too. These developments were essentially all carried out between 1950 and 1960. After that the
Atiyah–Singer index theorem opened another route to generalization. Consequently, the Euler characteristic of a
coherent sheaf is reasonably computable. For just one summand within the alternating sum, further arguments such as
vanishing theorems must be used.
See also
*
Arakelov theory
*
Grothendieck–Riemann–Roch theorem
*
Hirzebruch–Riemann–Roch theorem
*
Kawasaki's Riemann–Roch formula
*
Hilbert polynomial
*
Moduli of algebraic curves
Notes
References
*
*
* Grothendieck, Alexander, et al. (1966/67), Théorie des Intersections et Théorème de Riemann–Roch (SGA 6), LNM 225, Springer-Verlag, 1971.
*
* See pages 208–219 for the proof in the complex situation. Note that Jost uses slightly different notation.
* , contains the statement for curves over an algebraically closed field. See section IV.1.
*
* .
*
*
* ''Vector bundles on Compact Riemann Surfaces'', M. S. Narasimhan, pp. 5–6.
*
*
*
*
Misha KapovichThe Riemann–Roch Theorem(lecture note) an elementary introduction
* J. Gray
The ''Riemann–Roch theorem and Geometry, 1854–1914''.Is there a Riemann–Roch for smooth projective curves over an arbitrary field?on
MathOverflow
{{DEFAULTSORT:Riemann-Roch Theorem
Theorems in algebraic geometry
Geometry of divisors
Topological methods of algebraic geometry
Theorems in complex analysis
Bernhard Riemann