Hirzebruch–Riemann–Roch theorem

Builds on Projective variety Smooth morphism Vector bundle Sheaf cohomology Chern class Todd class

Statement

Let $X$ be a smooth projective variety of dimension $n$ over $\C$ , and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$ . Let $\chi(X, \mathcal{E}) = \sum_{i=0}^{n} (-1)^i \dim_\C H^i(X, \mathcal{E})$ , where $H^i$ is sheaf cohomology. Let $\alpha_1, \dots, \alpha_r$ be the Chern roots of $\mathcal{E}$ and $\beta_1, \dots, \beta_n$ those of the tangent bundle $T_X$ . So the Chern classes are the elementary symmetric functions in these roots. Let $\operatorname{ch}(\mathcal{E}) = \sum_{j=1}^{r} e^{\alpha_j}$ be the Chern character and $\operatorname{td}(T_X) = \prod_{j=1}^{n} \frac{\beta_j}{1 - e^{-\beta_j}}$ the Todd class, both in $H^{*}(X, \Q)$ . Then $\chi(X, \mathcal{E}) = \int_X \operatorname{ch}(\mathcal{E}) \, \operatorname{td}(T_X),$ where $\int_X$ takes the degree $2n$ component and evaluates it on the fundamental class.

Remarks

Hirzebruch proved the theorem with Thom's cobordism theory: both sides are multiplicative and compatible with the relevant operations, and they agree on enough generators. Later proofs come from the Grothendieck–Riemann–Roch theorem, applied to $X \to \operatorname{Spec} \C$ , and from the Atiyah–Singer index theorem, which extends it to all compact complex manifolds. The same formula holds for smooth projective varieties over any field, with the Chow ring in place of $H^*(X, \Q)$ .

Example: for a curve $X$ of genus $g$ and a line bundle $L$ of degree $d$ , $\operatorname{ch}(L) = 1 + c_1(L)$ and $\operatorname{td}(T_X) = 1 + \tfrac{1}{2} c_1(T_X)$ with $\deg c_1(T_X) = 2 - 2g$ . The degree $2$ part is $c_1(L) + \tfrac12 c_1(T_X)$ , so $\chi(L) = d + 1 - g$ , the classical Riemann–Roch theorem.

For a surface, the case $\mathcal{E} = \mathcal{O}_X$ gives Noether's formula $\chi(\mathcal{O}_X) = \tfrac{1}{12}(c_1^2 + c_2)$ .

Going deeper

For a student

In practice $\chi(\mathcal{E})$ is computed by the formula, and a vanishing theorem then kills all but one cohomology group, which gives the dimension of the space of sections. This is how one finds the dimension of linear systems on surfaces and higher-dimensional varieties. The theorem generalizes the Riemann–Roch theorem for curves and for surfaces, and it is in turn a special case of both the Grothendieck–Riemann–Roch theorem and the Atiyah–Singer index theorem.

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Chern class concept
Projective variety concept
Sheaf cohomology concept
Smooth morphism concept
Todd class concept
Vector bundle concept
Hirzebruch–Riemann–Roch theorem statement

Tag	`G3X02FKQCR`
URI	`m2://G3X02FKQCR`
Kind	statement, theorem
Writing	W0 stub
MSC	14C40, 14C17, 32L10
Status	This copy of the site has no build data. Run `m2 status G3X02FKQCR`.
About	Vector bundle, Chern class, Todd class
Uses	Projective variety, Smooth morphism, Vector bundle, Sheaf cohomology, Chern class, Todd class
Same as	Wikidata Q4663326 Wikipedia Hirzebruch–Riemann–Roch theorem
Changed	2 November 2026

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