Hirzebruch–Riemann–Roch theorem
Field | Algebraic geometry |
---|---|
First proof by | Friedrich Hirzebruch |
First proof in | 1954 |
Generalizations | Atiyah–Singer index theorem Grothendieck–Riemann–Roch theorem |
Consequences | Riemann–Roch theorem Riemann–Roch theorem for surfaces |
In
Statement of Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem applies to any holomorphic
of the dimensions as complex vector spaces, where n is the complex dimension of X.
Hirzebruch's theorem states that χ(X, E) is computable in terms of the Chern classes ck(E) of E, and the Todd classes of the holomorphic tangent bundle of X. These all lie in the cohomology ring of X; by use of the fundamental class (or, in other words, integration over X) we can obtain numbers from classes in The Hirzebruch formula asserts that
where the sum is taken over all relevant j (so 0 ≤ j ≤ n), using the
where is the Todd class of the tangent bundle of X.
Significant special cases are when E is a complex
Riemann Roch theorem for curves
For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classical
- (integrated over X).
But h0(O(D)) is just l(D), the dimension of the linear system of D, and by
For vector bundles V, the Chern character is rank(V) + c1(V), so we get Weil's Riemann Roch theorem for vector bundles over curves:
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.
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).
Asymptotic Riemann–Roch
Let D be an ample Cartier divisor on an irreducible projective variety X of dimension n. Then
More generally, if is any coherent sheaf on X then
See also
- Grothendieck–Riemann–Roch theorem - contains many computations and examples
- Hilbert polynomial- HRR can be used to compute Hilbert polynomials
References
- ISBN 3-540-58663-6