In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
. The classical form of it was first given by , after preliminary versions of it were found by and . The
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
-theoretic version is due to Hirzebruch.
Statement
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 In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the ex ...
, the dot . is the
intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
, 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 In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
Projective varieties
Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
of the surface. For comparison, the Riemann–Roch theorem for a curve states that χ(''D'') = χ(0) + deg(''D'').
Noether's formula
Noether's formula states that
:
where χ=χ(0) is the holomorphic Euler characteristic, ''c''
12 = (''K''.''K'') is a
Chern number
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
and the self-intersection number of the canonical class ''K'', and ''e'' = ''c''
2 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
In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebra ...
for surfaces.
Relation to the Hirzebruch–Riemann–Roch theorem
For surfaces, the
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 generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebra ...
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
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
''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
, and the Chern character of the sheaf ''L'' 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
:
(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
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
, and we get a more classical version of Riemann Roch for surfaces:
:
If we want, we can use
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexa ...
to express ''h''
2(O(''D'')) as ''h''
0(O(''K'' − ''D'')), but unlike the case of curves there is in general no easy way to write the ''h''
1(O(''D'')) term in a form not involving sheaf cohomology (although in practice it often vanishes).
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
*''r'' is the dimension of the complete linear system , ''D'', of a divisor ''D'' (so ''r'' = ''h''
0(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.D)/2
*''p''
''a'' is the arithmetic genus χ(O
''F'') − 1 of the surface
*''i'' is the index of speciality of ''D'', equal to dim ''H''
0(O(''K'' − ''D'')) (which by Serre duality is the same as dim ''H''
2(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 ''H''
1(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.
References
* ''Topological Methods in Algebraic Geometry'' by Friedrich Hirzebruch
*
*
{{DEFAULTSORT:Riemann-Roch theorem for surfaces
Theorems in algebraic geometry
Algebraic surfaces
Topological methods of algebraic geometry