Riemann–Roch-type theorem


In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous are Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al.

Formulation due to Baum, Fulton and MacPherson

Let and be functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that
Also, if is a local complete intersection morphism; i.e., it factors as a closed regular embedding into a smooth scheme P followed by a smooth morphism, then let
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:
between the two functors such that for each scheme X in C, the homomorphism satisfies: for a local complete intersection morphism, when there are closed embeddings into smooth schemes,
where refers to the Todd class.
Moreover, it has the properties:
Over the complex numbers, the theorem is a special case of the equivariant index theorem.

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.
The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for quotient stacks by finite groups.
One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class.