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 encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle.

The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem.

History

It is named for J. A. Todd, who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the Todd-Eger class. The general definition in higher dimensions is due to Friedrich Hirzebruch.

Definition

To define the Todd class $\operatorname {td} (E)$ where $E$ is a complex vector bundle on a topological space $X$ , it is usually possible to limit the definition to the case of a Whitney sum of line bundles, by means of a general device of characteristic class theory, the use of Chern roots (aka, the splitting principle). For the definition, let

Q(x)={\frac {x}{1-e^{-x}}}=1+{\dfrac {x}{2}}+\sum _{i=1}^{\infty }{\frac {B_{2i}}{(2i)!}}x^{2i}=1+{\dfrac {x}{2}}+{\dfrac {x^{2}}{12}}-{\dfrac {x^{4}}{720}}+\cdots

be the formal power series with the property that the coefficient of $x^{n}$ in $Q(x)^{n+1}$ is 1, where $B_{i}$ denotes the $i$ -th Bernoulli number. Consider the coefficient of $x^{j}$ in the product

\prod _{i=1}^{m}Q(\beta _{i}x)\

for any $m>j$ . This is symmetric in the $\beta _{i}$ s and homogeneous of weight $j$ : so can be expressed as a polynomial $\operatorname {td} _{j}(p_{1},\ldots ,p_{j})$ in the elementary symmetric functions $p$ of the $\beta _{i}$ s. Then $\operatorname {td} _{j}$ defines the Todd polynomials: they form a multiplicative sequence with $Q$ as characteristic power series.

If $E$ has the $\alpha _{i}$ as its Chern roots, then the Todd class

\operatorname {td} (E)=\prod Q(\alpha _{i})

which is to be computed in the cohomology ring of $X$ (or in its completion if one wants to consider infinite-dimensional manifolds).

The Todd class can be given explicitly as a formal power series in the Chern classes as follows:

\operatorname {td} (E)=1+{\frac {c_{1}}{2}}+{\frac {c_{1}^{2}+c_{2}}{12}}+{\frac {c_{1}c_{2}}{24}}+{\frac {-c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}}{720}}+\cdots

where the cohomology classes $c_{i}$ are the Chern classes of $E$ , and lie in the cohomology group $H^{2i}(X)$ . If $X$ is finite-dimensional then most terms vanish and $\operatorname {td} (E)$ is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

\operatorname {td} (E\oplus F)=\operatorname {td} (E)\cdot \operatorname {td} (F).

Let $\xi \in H^{2}({\mathbb {C} }P^{n})$ be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of ${\mathbb {C} }P^{n}$

0\to {\mathcal {O}}\to {\mathcal {O}}(1)^{n+1}\to T{\mathbb {C} }P^{n}\to 0,

one obtains ^[1]

\operatorname {td} (T{\mathbb {C} }P^{n})=\left({\dfrac {\xi }{1-e^{-\xi }}}\right)^{n+1}.

Computations of the Todd class

For any algebraic curve $C$ the Todd class is just $\operatorname {td} (C)=1+{\frac {1}{2}}c_{1}(T_{C})$ . Since $C$ is projective, it can be embedded into some $\mathbb {P} ^{n}$ and we can find $c_{1}(T_{C})$ using the normal sequence

$0\to T_{C}\to T_{\mathbb {P^{n}} }|_{C}\to N_{C/\mathbb {P} ^{n}}\to 0$

and properties of chern classes. For example, if we have a degree $d$ plane curve in $\mathbb {P} ^{2}$ , we find the total chern class is

${\begin{aligned}c(T_{C})&={\frac {c(T_{\mathbb {P} ^{2}}|_{C})}{c(N_{C/\mathbb {P} ^{2}})}}\\&={\frac {1+3[H]}{1+d[H]}}\\&=(1+3[H])(1-d[H])\\&=1+(3-d)[H]\end{aligned}}$