Riemann–Roch theorem on a surface

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 Castelnuovo (1896, 1897), after preliminary versions of it were found by Noether (1886) and Enriques (1894). The sheaf-theoretic version is due to Hirzebruch.

One form of the Riemann–Roch theorem states that if D is a divisor on a non-singular projective surface then

where χ is the holomorphic Euler characteristic, the dot . is the intersection number, and K is the canonical divisor. The constant χ(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + p_a, where p_a is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve states that χ(D) = χ(0) + deg(D).

Noether's formula states that

where χ=χ(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 χ(0) in the Riemann–Roch theorem with topological terms; this gives the Hirzebruch–Riemann–Roch theorem for surfaces.

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 is ${\displaystyle 1+c_{1}(X)/2+(c_{1}(X)^{2}+c_{2}(X))/12}$, and the Chern character of the sheaf L is just ${\displaystyle 1+c_{1}(L)+c_{1}(L)^{2}/2}$, so the Hirzebruch–Riemann–Roch theorem states that

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