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 $\operatorname(E)$ 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

::$Q(x) = \frac=1+\dfrac+\sum_^\infty \fracx^ = 1 +\dfrac+\dfrac-\dfrac+\cdots$
be the formal power series with the property that the coefficient of $x^n$ in $Q(x)^$ is 1, where $B_i$ denotes the $i$-th Bernoulli number. Consider the coefficient of $x^j$ in the product

:$\prod_^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 $\operatorname_j(p_1,\ldots, p_j)$ in the elementary symmetric functions $p$ of the $\beta_i$s. Then $\operatorname_j$ 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

:$\operatorname(E) = \prod Q(\alpha_i)$

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:

:$\operatorname(E) = 1 + \frac + \frac + \frac + \frac + \cdots$

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

Properties of the Todd class

The Todd class is multiplicative:
::$\operatorname(E\oplus F) = \operatorname(E)\cdot \operatorname(F).$

Let $\xi \in H^2( P^n)$ be the fundamental class of the hyperplane section.
From multiplicativity and the Euler exact sequence for the tangent bundle of $P^n$
:: $0 \to \to (1)^ \to T P^n \to 0,$
one obtains
Intersection Theory Class 18
by Ravi Vakil
:: $\operatorname(T P^n) = \left( \dfrac \right)^.$

Computations of the Todd class

For any algebraic curve $C$ the Todd class is just $\operatorname(X) = 1 + c_1(T_X)$. Since $C$ is projective, it can be embedded into some $\mathbb^n$ and we can find $c_1(T_X)$ using the normal sequence

$0 \to T_X \to T_\mathbb^n, _X \to N_ \to 0$

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

\begin
c(T_C) &= \frac \\
&= \frac \\
&= (1+3 (1-d \\
&= 1 + (3-d) \end

where /math> is the hyperplane class in $\mathbb^2$ restricted to $C$.

Hirzebruch-Riemann-Roch formula

For any coherent sheaf ''F'' on a smooth
compact complex manifold ''M'', one has
::$\chi(F)=\int_M \operatorname(F) \wedge \operatorname(TM),$
where $\chi(F)$ is its holomorphic Euler characteristic,
::$\chi(F):= \sum_^ (-1)^i \text_ H^i(M,F),$
and $\operatorname(F)$ its Chern character.

See also

* Genus of a multiplicative sequence

Notes

References

*
* Friedrich Hirzebruch, ''Topological methods in algebraic geometry'', Springer (1978)
*{{springer, id=T/t092930, title=Todd class, author=M.I. Voitsekhovskii

Characteristic classes