Inmathematics, theHirzebruch–Riemann–Roch theorem, named afterFriedrich Hirzebruch,Bernhard Riemann, andGustav Roch, is Hirzebruch's 1954 result generalizing the classicalRiemann–Roch theorem onRiemann surfaces to all complexalgebraic varieties of higher dimensions. The result paved the way for theGrothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.

The Hirzebruch–Riemann–Roch theorem applies to any holomorphicvector bundleE on acompactcomplex manifoldX, to calculate theholomorphic Euler characteristic ofE insheaf cohomology, namely the alternating sum^[1]

${\displaystyle \chi (X,E)=\sum _{i=0}^n(-1{)}^i{\dim }_{\mathbb{C}}{H}^i(X,E)}$

of the dimensions as complex vector spaces, wheren is the complex dimension ofX.

Hirzebruch's theorem states that χ(X,E) is computable in terms of theChern classesc_k(E) ofE, and theTodd classes${\displaystyle {\mathrm{td}}_j(X)}$ of the holomorphictangent bundle ofX. These all lie in thecohomology ring ofX; by use of thefundamental class (or, in other words, integration overX) we can obtain numbers from classes in${\displaystyle H^{2n}(X).}$ The Hirzebruch formula asserts that

${\displaystyle \chi (X,E)=\sum _{j=0}^n{\mathrm{ch}}_{n-j}(E){\mathrm{td}}_j(X),}$

using theChern character ch(E) in cohomology. In other words, the products are formed in the cohomology ring of all the 'matching' degrees that add up to 2n. Formulated differently, it gives the equality^[2]

${\displaystyle \chi (X,E)={\int }_X\mathrm{ch}(E)\mathrm{td}(X)}$

where${\displaystyle \mathrm{td}(X)}$ is theTodd class of the tangent bundle ofX.

Significant special cases are whenE is a complexline bundle, and whenX is analgebraic 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 theTodd classes are in some sense reciprocals of theChern Character.

For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classicalRiemann–Roch theorem. To see this, recall that for eachdivisorD on a curve there is aninvertible sheaf O(D) (which corresponds to a line bundle) such that thelinear system ofD is more or less the space of sections of O(D). For curves the Todd class is${\displaystyle 1+c_1(T(X))/2,}$ and the Chern character of a sheaf O(D) is just 1+c₁(O(D)), so the Hirzebruch–Riemann–Roch theorem states that

${\displaystyle h^0(\mathcal{O}(D))-h^1(\mathcal{O}(D))=c_1(\mathcal{O}(D))+c_1(T(X))/2}$ (integrated overX).

Buth⁰(O(D)) is justl(D), the dimension of the linear system ofD, and bySerre dualityh¹(O(D)) =h⁰(O(K − D)) =l(K − D) whereK is thecanonical divisor. Moreover,c₁(O(D)) integrated overX is the degree ofD, andc₁(T(X)) integrated overX is the Euler class 2 − 2g of the curveX, whereg is the genus. So we get the classical Riemann Roch theorem

${\displaystyle \ell (D)-\ell (K-D)=\text{deg}(D)+1-g.}$

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

${\displaystyle h^0(V)-h^1(V)=c_1(V)+\mathrm{rank}(V)(1-g).}$

For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially theRiemann–Roch theorem for surfaces

${\displaystyle \chi (D)=\chi (\mathcal{O})+((D.D)-(D.K))/2.}$

combined with the Noether formula.

If we want, we can use Serre duality to expressh²(O(D)) ash⁰(O(K − D)), but unlike the case of curves there is in general no easy way to write theh¹(O(D)) term in a form not involving sheaf cohomology (although in practice it often vanishes).

LetD be anampleCartier divisor on an irreducible projective varietyX of dimensionn. Then

${\displaystyle h^0\left(X,{\mathcal{O}}_X(mD)\right)=\frac{(D^n)}{n!}.m^n+O(m^{n-1}).}$

More generally, if${\displaystyle \mathcal{F}}$ is any coherent sheaf onX then

${\displaystyle h^0\left(X,\mathcal{F}\otimes {\mathcal{O}}_X(mD)\right)=\mathrm{rank}(\mathcal{F})\frac{(D^n)}{n!}.m^n+O(m^{n-1}).}$

• Grothendieck–Riemann–Roch theorem - contains many computations and examples
• Hilbert polynomial - HRR can be used to compute Hilbert polynomials

• Friedrich Hirzebruch,Topological Methods in Algebraic GeometryISBN 3-540-58663-6
• Huybrechts, Daniel (2004-11-18).Complex geometry: An introduction. Universitext.Springer Science+Business Media.ISBN 978-3540212904.{{cite book}}: CS1 maint: year (link)

• The Hirzebruch-Riemann-Roch Theorem

• Topological methods of algebraic geometry
• Theorems in complex geometry
• Theorems in algebraic geometry
• Bernhard Riemann

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• Short description matches Wikidata
• CS1 maint: year