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 algebraic varieties of higher dimensions.
The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf cohomology, namely the alternating sum 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
The Hirzebruch formula asserts that where the sum is taken over all relevant j (so 0 ≤ j ≤ n), using the Chern character ch(E) in cohomology.
Significant special cases are when E is a complex line bundle, and when X is an algebraic surface (Noether's formula).
The formula also expresses in a precise way the vague notion that the Todd classes are in some sense reciprocals of the Chern Character.
To see this, recall that for each divisor D on a curve there is an invertible sheaf O(D) (which corresponds to a line bundle) such that the linear system of D is more or less the space of sections of O(D).
and the Chern character of a sheaf O(D) is just 1+c1(O(D)), so the Hirzebruch–Riemann–Roch theorem states that But h0(O(D)) is just l(D), the dimension of the linear system of D, and by Serre duality h1(O(D)) = h0(O(K − D)) = l(K − D) where K is the canonical divisor.
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).