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 -functor in algebraic geometry" Russian Math. Surveys , 24 : 5 (1969) pp. 1–89 Uspekhi Mat. Nauk , 24 : 5 (1969) pp. 3–86 MR265355
[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 -theory and singular varieties" Acta Math. , 143 : 3–4 (1979) pp. 155–192
[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)