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 &mdash; 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:


 * $$\operatorname{td}(E) = 1 + \frac{c_1}{2} + \frac{c_1^2 +c_2}{12} + \frac{c_1c_2}{24} + \frac{-c_1^4 + 4 c_1^2 c_2 + c_1c_3 + 3c_2^2 - c_4}{720} + \cdots $$

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

Computations of the Todd class
For any algebraic curve $$C$$ the Todd class is just $$\operatorname{td}(C) = 1 + c_1(T_C)$$. 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"_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 $$\begin{align} c(T_C) &= \frac{c(T_{\mathbb{P}^2}|_C)}{c(N_{C/\mathbb{P}^2})} \\ &= \frac{1+3[H]}{1+d[H]} \\ &= (1+3[H])(1-d[H]) \\ &= 1 + (3-d)[H] \end{align}$$ where $$[H]$$ is the hyperplane class in $$\mathbb{P}^2$$ restricted to $$C$$.

Hirzebruch-Riemann-Roch formula
For any coherent sheaf F on a smooth compact complex manifold M, one has
 * $$\chi(F)=\int_M \operatorname{ch}(F) \wedge \operatorname{td}(TM),$$

where $$\chi(F)$$ is its holomorphic Euler characteristic,
 * $$\chi(F):= \sum_{i=0}^{\text{dim}_{\mathbb{C}} M} (-1)^i \text{dim}_{\mathbb{C}} H^i(M,F),$$

and $$\operatorname{ch}(F)$$ its Chern character.