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.
formula states that where χ=χ is the holomorphic Euler characteristic, c12 = is a Chern number and the self-intersection number of the canonical classK, and e = c2 is the topological Euler characteristic. It can be used to replace the term χ 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 sheafL = O such that the linear system of D is more or less the space of sections of L. For surfaces the Todd class is, and the Chern character of the sheafL is just, 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 For invertible sheaves 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 as h0, but unlike the case of curves there is in general no easy way to write the h1 term in a form not involving sheaf cohomology.
Early versions
The earliest forms of the Riemann-Roch 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
n is the virtual degree of D, given by the self-intersection number
π is the virtual genus of D, equal to 1 + /2
pa is the arithmetic genus χ − 1 of the surface
i is the index of speciality of D, equal to dim H0.
The difference between the two sides of this inequality was called the superabundances of the divisor D. Comparing this inequality with the sheaf-theoretic version of the Riemann-Roch theorem shows that the superabundance of D is given by s = dim H1. The divisor D was called regular if i = s = 0 and superabundant if s > 0.
