Riemann-Roch theorem
A theorem expressing the Euler characteristic of a locally free sheaf
on an algebraic or analytic variety
in terms of the characteristic Chern classes of
and
(cf. Chern class). It can be used to calculate the dimension of the space of sections of
(the Riemann–Roch problem).
The classical Riemann–Roch theorem relates to the case of non-singular algebraic curves and states that for any divisor
on
,
![]() | (1) |
where is the dimension of the space of functions
for which
,
is the canonical divisor and
is the genus of
. In the middle of the 19th century B. Riemann used analytic methods to obtain the inequality
![]() |
The equality (1) was proved by E. Roch.
The Riemann–Roch theorem for curves is the one-dimensional case of the more general Riemann–Roch–Hirzebruch–Grothendieck theorem. Let be a non-singular projective variety of dimension
, and let
be an appropriate cohomology theory: either
are singular cohomology spaces when the basic field
, or
where
is a Chow ring, or
is the ring associated to the Grothendieck ring
(see [2], [7]). Let
be a locally free sheaf of rank
on
. Universal polynomials for
with rational coefficients,
and
, in the Chern classes
of
are defined in the following way. The factorization
![]() |
is examined for the Chern polynomial, where the are formal symbols. The exponential Chern character is defined by the formula
![]() |
and, accordingly, the Todd class is defined as
![]() |
and
are symmetric functions in the
and they can be written as polynomials in
.
The Riemann–Roch–Hirzebruch theorem: If is a non-singular projective variety or a compact complex variety of dimension
and if
is a vector bundle of rank
on
, then
![]() |
where is the tangent sheaf on
and
denotes the component of degree
in
. This theorem was proved by F. Hirzebruch in the case of the ground field
. When
and the invertible sheaf
, it leads to the equation
![]() |
where is the second Chern class of the surface
and
is its canonical class. In particular, when
Noether's formula is obtained:
![]() |
For three-dimensional varieties the theorem leads to
![]() |
In particular, when ,
![]() |
In 1957, A. Grothendieck generalized the Riemann–Roch–Hirzebruch theorem to the case of a morphism of non-singular varieties over an arbitrary algebraically closed field (see [1]). Let and
be the Grothendieck groups of the coherent and locally free sheafs on
, respectively (cf. Grothendieck group). The functor
is a covariant functor from the category of schemes and proper morphisms into the category of Abelian groups. In this case, for a proper morphism
the morphism
is defined by the formula
![]() |
where is an arbitrary coherent sheaf on
and
is a covariant functor into the category of rings. For regular schemes with an ample sheaf, the groups
and
coincide and are denoted by
. The Chern character
is a homomorphism of rings;
is also a covariant functor: The Gizin homomorphism
is defined. When
, the homomorphism
is obtained from
for homology spaces using Poincaré duality. The theorem as generalized by Grothendieck expresses the measure of deviation from commutativity of the homomorphisms
and
.
The Riemann–Roch–Hirzebruch–Grothendieck theorem: Let be a smooth projective morphism of non-singular projective varieties; then for any
the equation
![]() |
in is true, where
(the relative tangent sheaf of the morphism
).
When is a point, this theorem reduces to the Riemann–Roch–Hirzebruch theorem. There are generalizations (see [5], [6], [7]) when
is a Noetherian scheme with an ample invertible sheaf, when
is a projective morphism whose fibres are locally complete intersections, and also to the case of singular quasi-projective varieties.
Several versions of the Riemann–Roch theorem are closely connected with the index problem for elliptic operators (see Index formulas). For example, the Riemann–Roch–Hirzebruch theorem for compact complex varieties is a particular case of the Atiyah–Singer index theorem.
References
[1] | A. Borel, J.-P. Serre, "La théorème de Riemann–Roch" Bull. Soc. Math. France , 86 (1958) pp. 97–136 |
[2] | Yu.I. Manin, "Lectures on the ![]() |
[3] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
[4] | F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) MR1335917 MR0202713 Zbl 0376.14001 |
[5] | P. Baum, W. Fulton, R. MacPherson, "Riemann–Roch for singular varieties" Publ. Math. IHES , 45 (1975) pp. 101–145 |
[6] | P. Baum, W. Fulton, R. MacPherson, "Riemann–Roch for topological ![]() |
[7] | "Théorie des intersections et théorème de Riemann–Roch" P. Berthelot (ed.) et al. (ed.) , Sem. Geom. Alg. 6 , Lect. notes in math. , 225 , Springer (1971) |
Comments
There are analogues of the Riemann–Roch theorem in algebraic number theory and arithmetic algebraic geometry.
References
[a1] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) MR0282947 Zbl 0211.38404 |
[a2] | K. Szpiro, "Sem. sur les pinceaux arithmétiques: La conjecture de Mordell" Astérisque , 127 (1985) |
Riemann-Roch theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann-Roch_theorem&oldid=22983