Hirzebruch-Riemann-Roch theorem

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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 was the first successful generalisation of the classical Riemann-Roch theorem on Riemann surfaces to all higher dimensions, and paved the way to the Grothendieck-Hirzebruch-Riemann-Roch theorem proved about three years later.

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[edit] Statement of Hirzebruch-Riemann-Roch theorem

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

\chi(X,E) = \dim H^{0}(X,E)- \dim H^{1}(X,E) + \dim H^{2} (X,E) - \cdots

of the dimensions as complex vector spaces. (By basic results on coherent cohomology these dimensions are all finite, and are 0 except for the first 2n + 1 cases, where X has complex dimension n, so the sum is finite.)

Hirzebruch's theorem states that χ(X, E) is computable in terms of the Chern classes Cj(E) of E, and the Todd polynomials Tj in the Chern 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 H2n(X). The Hirzebruch formula is the sum

\sum ch_{n-j}(E) \frac{T_{j}}{j!}

taken over all relevant j (so 0 ≤ j ≤ n), using the Chern character ch(E) in cohomology. In other words, the cross products are formed in cohomology ring of all the 'matching' degrees that add up to 2n, where to 'massage' the Cj(E) a formal manipulation is done, setting

ch(E) = \sum exp(x_{i})

and the total Chern class

C(E) = \sum C_{j}(E) = \prod (1 + x_{i}) .

Significant special cases are when E is a complex line bundle, and when X is an algebraic surface (Noether's formula). Weil's Riemann-Roch theorem for vector bundles on curves, and the Riemann-Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the Todd classes are in some sense reciprocals of characteristic classes.

[edit] Riemann Roch theorem for curves

For curves, the Hirzebruch-Riemann-Roch theorem is essentially the classical Riemann-Roch theorem. To see this, recall that for each divisor D on a curve there is a line bundle (or sheaf) O(D) such that the linear system of D is more or less the space of sections of O(D). For curves the Todd class is 1 + c1(T(X))/2, and the Chern character of a sheaf O(D) is just 1+c1(O(D)), so the Hirzebruch-Riemann-Roch theorem states that

h0(O(D)) − h1(O(D)) = c1(O(D)) +c1(T(X))/2 (integrated over X)

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. Moreover c1(O(D)) integrated over X is the degree of D, and c1(T(X)) integrated over X is the Euler class 2-2g of the curve X, where g is the genus. So we get the classical Riemann Roch theorem

l(D) - l(K-D) = deg(D) + 1 - g.

For vector bundles V, the Chern character is dim(V) + c1(V), so we get Weil's Riemann Roch theorem for vector bundles over curves:

h0(V) − h1(V) = c1(V) + dim(V)(1-g)

[edit] 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 a line bundle (or sheaf) O(D) such that the linear system of D is more or less the space of sections of O(D). For surfaces the Todd class is 1 +c1(T(X))/2 + (c1(T(X))2 + c2(T(X)))/12, and the Chern character of a sheaf O(D) is just 1+c1(O(D)) + (c1(O(D))2−c2(O(D)))/2, so the Hirzebruch-Riemann-Roch theorem states that

χ(D)) = h0(O(D)) − h1(O(D))+ h2(O(D))= (c1(O(D))2−c2(O(D)))/2 + c1(O(D))c1(T(X))/2 +(c1(T(X))2 + c2(T(X)))/12 (integrated over X)

Fortunately this can be written in a clearer form as follows. First putting D = 0 shows that

χ = h0(O(0)) − h1(O(0))+ h2(O(0)) = (c1(T(X))2 + c2(T(X)))/12 (Noether's formula).

For 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:

χ(D) = χ(0) + ((D.D)−(D.K))/2

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).

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