- Riemann–Roch theorem for surfaces
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, (,) 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 + pa, where pa 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.
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 is 1 + c1(X) / 2 + (c1(X)2 + c2(X)) / 12, and the Chern character of the sheaf L is just 1 + c1(L) + c1(L)2 / 2, 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
- (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(K − D)), 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).
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 Zariski (1995, p. 78), which states that
- 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)
- π is the virtual genus of D, equal to 1 + (D.D + K)/2
- pa is the arithmetic genus χ(OF) − 1 of the surface
- i is the index of speciality of D, equal to dim H0(O(K − D)) (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 Riemann–Roch 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 if s > 0.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Riemann–Roch theorem — In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates… … Wikipedia
Hirzebruch–Riemann–Roch theorem — In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch s 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions. It… … Wikipedia
Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… … Wikipedia
De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… … Wikipedia
Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… … Wikipedia
Bernhard Riemann — Infobox Scientist name =Bernhard Riemann box width =300px image width =225px caption =Bernhard Riemann, 1863 birth date =September 17, 1826 birth place =Breselenz, Germany death date =death date and age|1866|7|20|1826|9|17 death place =Selasca,… … Wikipedia
Gauss–Bonnet theorem — The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). It is named … Wikipedia
Gustav Roch — (december 9 1839 november 21 1866) was a German mathematician who made significant contributions to the theory of Riemann surfaces in a career that was prematurely curtailed at the age of 26.BiographyBorn in Leipzig, Roch attended the Polytechnic … Wikipedia
Max Noether's theorem — In mathematics, Max Noether s theorem in algebraic geometry may refer to at least six results of Max Noether. Noether s theorem usually refers to a result derived from work of his daughter Emmy Noether. There are several closely related results… … Wikipedia
Géométrie différentielle des surfaces — En mathématiques, la géométrie différentielle des surfaces est la branche de la géométrie différentielle qui traite des surfaces (les objets géométriques de l espace usuel E3, ou leur généralisation que sont les variétés de dimension 2), munies… … Wikipédia en Français