In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by , after preliminary versions of it were found by and . The sheaf-theoretic version is due to Hirzebruch.

Statement

One form of the Riemann is a divisor on a non-singular projective surface then

is the holomorphic Euler characteristic is the intersection number is the arithmetic genus of the surface. For comparison, the Riemann.

Noether's formula

Noether's formula states that

where =(0) is the holomorphic Euler characteristic, c12 =(K.K) is a Chern number and the self-intersection number of the canonical class K, and e=c2 is the topological Euler characteristic. It can be used to replace the term (0) in the Riemann–Roch theorem with topological terms; this gives the Hirzebruch–Riemann–Roch theorem for surfaces.

Relation to the Hirzebruch–Riemann–Roch theorem

For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces combined with the Noether formula. To see this, recall that for each divisor D on a surface there is an invertible sheaf L = O(D) such that the linear system of D is more or less the space of sections of L. For surfaces the Todd class, and the Chern character of the sheaf L, so the Hirzebruch–Riemann–Roch theorem states that

Fortunately this can be written in a clearer form as follows. First putting D=0 shows that

1. (Noether's formula)

For invertible sheaves (line bundles) the second Chern class vanishes. The products of second cohomology classes can be identified with intersection numbers in the Picard group, and we get a more classical version of Riemann Roch for surfaces:

If we want, we can use Serre duality to express h2(O(D)) as h0(O(KD)), but unlike the case of curves there is in general no easy way to write the h1(O(D)) term in a form not involving sheaf cohomology (although in practice it often vanishes).

Early versions

The earliest forms of the RiemannRoch theorem for surfaces were often stated as an inequality rather than an equality, because there was no direct geometric description of first cohomology groups. A typical example is given by , which states that

where

• r is the dimension of the complete linear system |D| of a divisor D (so r=h0(O(D)) 1)

• n is the virtual degree of D, given by the self-intersection number (D.D)

• is the virtual genus of D, equal to 1 + (D.D + K.D)/2

• pa is the arithmetic genus (OF) 1 of the surface

• i is the index of speciality of D, equal to dim H0(O(KD)) (which by Serre duality is the same as dim H2(O(D))).

The difference between the two sides of this inequality was called the superabundance s of the divisor D. Comparing this inequality with the sheaf-theoretic version of the RiemannRoch theorem shows that the superabundance of D is given by s=dim H1(O(D)). The divisor D was called regular if i=s=0 (or in other words if all higher cohomology groups of O(D) vanish) and superabundant ifs>0.

References

• Topological Methods in Algebraic Geometry by Friedrich Hirzebruch