In algebraic geometry, there are various generalizations of the

Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...

; among the most famous is the

Grothendieck–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 it ...

, which is further generalized by the formulation due to Fulton et al.

Formulation due to Baum, Fulton and MacPherson

Let

G_*

and

A_*

be functors on the category ''C'' of schemes separated and locally of finite type over the base field ''k'' with

proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field '' ...

s such that *

G_*(X)

is the

Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...

coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...

on ''X'', *

A_*(X)

is the rational

Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...

of ''X'', *for each proper morphism ''f'',

G_*(f), A_*(f)

are the direct images (or push-forwards) along ''f''. Also, if

f: X \to Y

is a (global) local complete intersection morphism; i.e., it factors as a closed regular embedding

X \hookrightarrow P

into a smooth scheme ''P'' followed by a smooth morphism

P \to Y

, then let :

T_f =_X -_/math>
be the class in the Grothendieck group of vector bundles on ''X''; it is independent of the factorization and is called the virtual tangent bundle of ''f''.

Then the Riemann–Roch theorem amounts to the construction of a unique

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

: :

\tau: G_* \to A_*

between the two functors such that for each scheme ''X'' in ''C'', the homomorphism

\tau_X : G(X) \to A(X)

satisfies: for a local complete intersection morphism

f: X \to Y

, when there are closed embeddings

X \subset M, Y \subset P

into smooth schemes, :

\tau_X f^* = \operatorname(T_f) \cdot f^* \tau_Y

where

\operatorname

refers to the

Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounter ...

. Moreover, it has the properties: *

\tau_X(\beta \otimes \alpha) = \operatorname(\beta) \tau(\alpha)

for each

\alpha \in G_*(X)

and the

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

\operatorname(\beta)

(or the action of it) of the

\beta

in the Grothendieck group of vector bundles on ''X''. *it ''X'' is a closed subscheme of a smooth scheme ''M'', then the theorem is (roughly) the restriction of the theorem in the smooth case and can be written down in terms of a

localized Chern class In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's ''intersection theory'', as an algeb ...

The equivariant Riemann–Roch theorem

Over the complex numbers, the theorem is (or can be interpreted as) a special case of the

equivariant index theorem In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element. I ...

The Riemann–Roch theorem for Deligne–Mumford stacks

Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case ( Kawasaki's Riemann–Roch). The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for

quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. T ...

s by finite groups. One of the significant applications of the theorem is that it allows one to define a

virtual fundamental class In mathematics, specifically enumerative geometry, the virtual fundamental class \text_ of a space X is a replacement of the classical fundamental class \in A^*(X) in its chow ring which has better behavior with respect to the enumerative probl ...

in terms of the ''K''-theoretic virtual fundamental class.

Notes

References

* * * * * *Vakil
Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry

External links