Grothendieck–Hirzebruch–Riemann–Roch theorem

In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Hirzebruch–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.

The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Hirzebruch–Riemann–Roch theorem can be proved using the families index theorem. Alexander Grothendieck, its author, was rumored to have finished the proof around 1956 but did not publish his theorem because he was not satisfied with it. Instead Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's preliminary (as he saw it) proof.

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Formulation

Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group

K_0(X)\,

of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation

\mbox{ch} \colon K_0(X) \to A(X, {\Bbb Q})

where

A_d(X,{\Bbb Q})\,

is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group

H^{2 \mathrm{dim}(X) - 2d}(X, {\Bbb Q}).

Now consider a proper morphism

f \colon X \to Y\,

between smooth quasi-projective schemes and a bounded complex of sheaves {\mathcal F^\bull}.

The Grothendieck–Riemann–Roch theorem relates the push forward maps

f_{\mbox{!}} = \sum (-1)^i R^i f_* \colon K_0(X) \to K_0(Y)

and the pushforward

f_* \colon A(X) \to A(Y),\,

by the formula

\mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull)\mbox{td}(Y) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(X) ).

Here td(X) is the Todd genus of (the tangent bundle of) X. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depends on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck Hirzebruch Riemann Roch formula to

\mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(T_f) )

where T_f is the relative tangent sheaf of f. This is often useful in applications, for example if f is a locally trivial fibration.

Generalising and specialising

Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch(—)td(X) and to the non-proper case by considering cohomology with compact support.

The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes.

The Hirzebruch–Riemann–Roch theorem is (essentially) the special case where Y is a point and the field is the field of complex numbers.

History

Alexander Grothendieck's version of the Riemann–Roch theorem was originally conveyed in a letter to Jean-Pierre Serre around 1956–7. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel–Serre exposition.

The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties. In short, he applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K-theory.

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