Chern classes are an ubiquitous concept in mathematics, being part of algebraic geometry, algebraic topology, and differential geometry. In this post we discuss Chern classes in the context of algebraic geometry, where they are part of intersection theory (see Algebraic Cycles and Intersection Theory). Among the applications of the theory of Chern classes is a higher-dimensional generalization of the Riemann-Roch theorem (see More on Sheaves) called the Hirzebruch-Riemann-Roch theorem. There is an even further generalization called the Grothendieck-Riemann-Roch theorem, which concerns a morphism of nonsingular projective varieties , and for which the Hirzebruch-Riemann-Roch theorem is merely the case where is a point.
Let be a nonsingular projective variety, and let be the Chow ring of (see Algebraic Cycles and Intersection Theory). Let be a locally free sheaf of rank on .
We recall that locally free sheaves correspond to vector bundles (see Vector Fields, Vector Bundles, and Fiber Bundles and More on Sheaves). Therefore, their fibers are isomorphic to . The projective bundle associated to the locally free sheaf is essentially obtained by replacing the fibers with projective spaces (see Projective Geometry).
Let be the class of the divisor corresponding to the twisting sheaf (see More on Sheaves) . Let be the projection of the fiber bundle to its “base space” . Then the pullback makes into a free module generated by , , .
We define the -th Chern class by the requirement that and
where the dot denotes the intersection product (see Algebraic Cycles and Intersection Theory).
Chern classes are associated to locally free sheaves, which, as we have already mentioned, correspond to vector bundles, and are elements of the Chow ring. We can therefore think of them as generalizing the correspondence between line bundles (vector bundles of dimension ) and elements of the Picard group, since, as mentioned in Algebraic Cycles and Intersection Theory, the Chow ring is kind of an analogue of the Picard group for higher dimensions.
We can also define the total Chern class
and the Chern polynomial
Chern classes satisfy the following important properties:
(i) If is the line bundle associated to a divisor , then .
(ii) If is a morphism, and is a locally free sheaf on , then for each ,
(iii) If is an exact sequence (see Exact Sequences) of locally free sheaves, then
These three properties can also be considered as a set of axioms which define the Chern classes, instead of the definition that we gave earlier.
Another important property of Chern classes, which comes from the so-called splitting principle, allows us to factor the Chern polynomial into the Chern polynomials of line bundles, and so we have:
The are called the Chern roots of .
We define the exponential Chern character (or simply Chern character) as
and the Todd class as
Now we can discuss the generalizations of the Riemann-Roch theorem. We first review the statement of the Riemann-Roch theorem for curves, but we restate it slightly in terms of the Euler characteristic.
The Euler characteristic of a coherent sheaf on a projective scheme over a field is defined to be the alternating sum of the dimensions of the cohomology groups (see Cohomology in Algebraic Geometry) as vector spaces over .
Then we can state the Riemann-Roch theorem for curves as
The connection of this formulation with the one we gave in More on Sheaves, where the left-hand side is given by is provided by the fact that is the same as (and in fact defined as) , together with the theorem known as Serre duality, which says that is dual to , where denotes the dual of the line bundle .
The Hirzebruch-Riemann-Roch theorem says that
where is the tangent bundle of (the dual of the cotangent bundle of , as defined in More on Sheaves) and is the component of degree in .
Finally we come to the even more general Grothendieck-Riemann-Roch theorem, but first we must introduce the Grothendieck group of a scheme , which eventually inspired the area of mathematics known as K-theory.
The Grothendieck group of a scheme is defined to be the quotient of the free abelian group generated by the coherent sheaves on by the subgroup generated by expressions of the form whenever there is an exact sequence
of coherent sheaves on . Intuitively, we may think of the Grothendieck group as follows. The isomorphism classes of vector bundles on form a commutative monoid under the operation of taking the direct sum of vector bundles (also called the Whitney sum). There is a way to obtain an abelian group from this monoid, called the group completion, and the abelian group we obtain is the Grothendieck group. The Chern classes and the Chern character are also defined on the Grothendieck group . In K-theory, the Grothendieck group is also denoted .
If is a proper morphism (a morphism that is separable, of finite type, and universally closed, i.e. for every scheme , the projection maps closed sets to closed sets), we have a map defined by
where the are the higher direct image functors, which are defined as the right derived functors (The Hom and Tensor Functors) of the direct image functor (see Direct Images and Inverse Images of Sheaves).
The Grothendieck-Riemann-Roch theorem says that for any , we have
The Grothendieck-Riemann-Roch theorem is one of the most general versions of the Riemann-Roch theorem, a classic theorem whose origins date back to the 19th century. However, there are also other generalizations, such as the arithmetic Riemann-Roch theorem which is closely related to number theory, and the Atiyah-Singer index theorem which is closely related to physics. We leave these, and the many other details of the topics we have discussed in this post (along with the theory of Chern classes in the context of algebraic topology and differential geometry), to the references for now, until we can discuss them on this blog in the future.
The featured image on this post is a handwritten comment of Alexander Grothendieck, apparently from a lecture in 1971, featuring the Grothendieck-Riemann-Roch theorem.
Algebraic Geometry by Robin Hartshorne