Todd class

Short description: Characteristic class in algebraic topology

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 $\mathrm{td}(E)$ 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

$Q(x)=\frac{x}{1-e^{-x}}=\sum _{i=0}^{\mathrm{\infty }}\frac{B_i}{i!}{x}^i=1+\frac{x}{2}+\frac{x^2}{12}-\frac{x^4}{720}+\cdots$

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 (with $B_1=+\frac{1}{2}$). 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 ${\mathrm{td}}_j(p_1,\dots ,p_j)$ in the elementary symmetric functions $p$ of the ${\beta }_i$s. Then ${\mathrm{td}}_j$ 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

$\mathrm{td}(E)=\prod Q({\alpha }_i)$

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:

$\mathrm{td}(E)=1+\frac{c_1}{2}+\frac{c_1^2+c_2}{12}+\frac{c_1{c}_2}{24}+\frac{-c_1^4+4c_1^2{c}_2+c_1{c}_3+3c_2^2-c_4}{720}+\cdots$

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 $\mathrm{td}(E)$ is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

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

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 $\mathbb{ℂ}{P}^n$

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

one obtains

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

Computations of the Todd class

For any algebraic curve

$C$

the Todd class is just

$\mathrm{td}(C)=1+\frac{1}{2}{c}_1(T_C)$

. Since

$C$

is projective, it can be embedded into some

${\mathbb{\mathbb{P}}}^n$

and we can find

$c_1(T_C)$

using the normal sequence

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

and properties of chern classes. For example, if we have a degree

$d$

plane curve in

${\mathbb{\mathbb{P}}}^2$

, we find the total chern class is

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

where

$[H]$

is the hyperplane class in

${\mathbb{\mathbb{P}}}^2$

restricted to

$C$

.

Hirzebruch-Riemann-Roch formula

Main article: Hirzebruch–Riemann–Roch theorem

For any coherent sheaf F on a smooth compact complex manifold M, one has

$\chi (F)=\underset{M}{\int }\mathrm{ch}(F)\wedge \mathrm{td}(TM),$

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

See also

• Genus of a multiplicative sequence

References

• Todd, J. A. (1937), "The Arithmetical Invariants of Algebraic Loci", Proceedings of the London Mathematical Society 43 (1): 190–225, doi:10.1112/plms/s2-43.3.190 
• Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
• Hazewinkel, Michiel, ed. (2001), "Todd class", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=T/t092930 
• Huybrechts, Daniel (2004-11-18) (in en). Complex geometry: An introduction. Universitext. Springer Science+Business Media. ISBN 978-3540212904. https://link.springer.com/book/10.1007/b137952.