Chern Classes and Generalized Riemann-Roch Theorems

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 $f:X\rightarrow Y$ , and for which the Hirzebruch-Riemann-Roch theorem is merely the case where $Y$ is a point.

Let $X$ be a nonsingular projective variety, and let $A(X)$ be the Chow ring of $X$ (see Algebraic Cycles and Intersection Theory). Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on $X$ .

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 $\mathbb{A}^{r}$ . The projective bundle $\mathbb{P}(\mathcal{E})$ associated to the locally free sheaf $\mathcal{E}$ is essentially obtained by replacing the fibers with projective spaces $\mathbb{A}\setminus\{0\}/k^{*}$ (see Projective Geometry).

Let $\xi\in A^{1}(\mathbb{P}(\mathcal{E}))$ be the class of the divisor corresponding to the twisting sheaf (see More on Sheaves) $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ . Let $\pi:\mathbb{P}(\mathcal{E})\rightarrow X$ be the projection of the fiber bundle $\mathbb{P}(\mathcal{E})$ to its “base space” $X$ . Then the pullback $\pi^{*}:A^{i}(X)\rightarrow A^{i+r-1}(\mathbb{P}(\mathcal{E}))$ makes $A(\mathbb{P}(\mathcal{E}))$ into a free $A(X)$ module generated by $1$ , $\xi$ , $\xi^{2},...,\xi^{r-1}$ .

We define the $i$ -th Chern class $c_{i}(\mathcal{E})\in A^{i}(X)$ by the requirement that $c_{0}(\mathcal{E})=1$ and

$\displaystyle \sum_{i=0}^{r}(-1)^{i}\pi^{*}c_{i}(\mathcal{E}).\xi^{r-i}=0$

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 $1$ ) 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

$\displaystyle c(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})...+c_{r}(\mathcal{E})$

and the Chern polynomial

$\displaystyle c_{t}(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})t+...+c_{r}(\mathcal{E})t^{r}$ .

Chern classes satisfy the following important properties:

(i) If $\mathcal{E}$ is the line bundle $\mathcal{L}(D)$ associated to a divisor $D$ , then $c_{t}=1+Dt$ .

(ii) If $f:X'\rightarrow X$ is a morphism, and $\mathcal{E}$ is a locally free sheaf on $X$ , then for each $i$ ,

$\displaystyle c_{i}(f^{*}\mathcal{E})=f^{*}c_{i}(\mathcal{E})$ .

(iii) If $0\rightarrow\mathcal{E}'\rightarrow\mathcal{E}\rightarrow\mathcal{E}''\rightarrow 0$ is an exact sequence (see Exact Sequences) of locally free sheaves, then

$\displaystyle c_{t}(\mathcal{E})=c_{t}(\mathcal{E}')\cdot c_{t}(\mathcal{E}'')$

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:

$c_{t}(\mathcal{E})=\prod_{i=1}^{r}(1+a_{i}t)$

The $a_{i}$ are called the Chern roots of $\mathcal{E}$ .

We define the exponential Chern character (or simply Chern character) as

$\displaystyle \text{ch}(\mathcal{E})=\sum_{i=1}^{r}e^{a_{i}}$

and the Todd class as

$\displaystyle \text{td}(\mathcal{E})=\prod_{i=1}^{r}\frac{(a_{i})}{1-e^{-a_{i}}}$ .

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 $\mathcal{E}$ on a projective scheme $X$ over a field $k$ is defined to be the alternating sum of the dimensions of the cohomology groups $H^{i}(X,\mathcal{F})$ (see Cohomology in Algebraic Geometry) as vector spaces over $k$ .

$\displaystyle \chi(\mathcal{E})=\sum_{i}(-1)^{i}\text{dim}_{k}H^{i}(X,\mathcal{F})$ .

Then we can state the Riemann-Roch theorem for curves as

$\chi(\mathcal{L}(D))=\text{deg}(D)+1-g$ .

The connection of this formulation with the one we gave in More on Sheaves, where the left-hand side is given by $h^{0}(D)-h^{0}(K_{X}-D)$ is provided by the fact that $h^{0}(D)$ is the same as (and in fact defined as) $\text{dim}_{k}H^{0}(X, \mathcal{L}(D))$ , together with the theorem known as Serre duality, which says that $H^{1}(X,\mathcal{L}(D))$ is dual to $H^{0}(X,\omega\otimes\mathcal{L}(D)^{\vee})$ , where $\mathcal{L}(D)^{\vee}$ denotes the dual of the line bundle $\mathcal{L}(D)$ .

The Hirzebruch-Riemann-Roch theorem says that

$\displaystyle \chi(\mathcal{E})=\text{deg}(\text{ch}(\mathcal{E}).\text{td}(\mathcal{T}_{X}))_{n}$

where $\mathcal{T}_{X}$ is the tangent bundle of $X$ (the dual of the cotangent bundle of $X$ , as defined in More on Sheaves) and $(\quad)_{n}$ is the component of degree $n$ in $A(X)\otimes\mathbb{Q}$ .

Finally we come to the even more general Grothendieck-Riemann-Roch theorem, but first we must introduce the Grothendieck group $K(X)$ of a scheme $X$ , which eventually inspired the area of mathematics known as K-theory.

The Grothendieck group $K(X)$ of a scheme $X$ is defined to be the quotient of the free abelian group generated by the coherent sheaves on $X$ by the subgroup generated by expressions of the form $\mathcal{F}-\mathcal{F}'-\mathcal{F}''$ whenever there is an exact sequence

$\displaystyle 0\rightarrow\mathcal{F'}\rightarrow\mathcal{F}\rightarrow\mathcal{F''}\rightarrow 0$

of coherent sheaves on $X$ . Intuitively, we may think of the Grothendieck group as follows. The isomorphism classes of vector bundles on $X$ 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 $K(X)$ . In K-theory, the Grothendieck group $K(X)$ is also denoted $K_{0}(X)$ .

If $f:X\rightarrow Y$ is a proper morphism (a morphism that is separable, of finite type, and universally closed, i.e. for every scheme $Z\rightarrow Y$ , the projection $X\times_{Y}Z\rightarrow Z$ maps closed sets to closed sets), we have a map $f_{!}:K(X)\rightarrow Y$ defined by

$\displaystyle f_{!}(\mathcal{F})=\sum_{i}(-1)^{i}R^{i}f_{*}(\mathcal{F})$

where the $R^{i}f_{*}$ are the higher direct image functors, which are defined as the right derived functors (The Hom and Tensor Functors) of the direct image functor $f_{*}$ (see Direct Images and Inverse Images of Sheaves).

The Grothendieck-Riemann-Roch theorem says that for any $x\in K(X)$ , we have

$\displaystyle f_{*}(\text{ch}(x).\text{td}(\mathcal{T}_{X})=\text{ch}(f_{!}(x)).\text{td}(\mathcal{T}_{Y})$ .

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.

References:

Chern Class on Wikipedia

Projective Bundle on Wikipedia

Hirzebruch-Riemann-Roch Theorem on Wikipedia

Grothendieck-Riemann-Roch Theorem on Wikipedia

Chern Classes: Part 1 on Rigorous Trivialities

Chern Classes: Part 2 on Rigorous Trivialities

The Chow Ring and Chern Classes on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, Stated on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, a Proof-Sketch on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne