Todd class

In mathematics, the Todd class is a construction in the algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined in terms of Chern classes within various theories, notably in differential topology, 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 crucial 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 $td (E)$ 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, consider the formal power series

Q (x) = \frac{x}{1 - e^{- x}} = \sum_{i = 0}^{\infty} \frac{B_{i}}{i!} x^{i} = 1 + \frac{x}{2} + \frac{x^{2}}{12} - \frac{x^{4}}{720} + \dots

where $B_{i}$ denotes the $i$ -th Bernoulli number (with $B_{1} = \frac{1}{2}$ ). The coefficient of $x^{n}$ in $Q (x)^{n + 1}$ is 1, which follows from the Lagrange inversion formula applied to the inverse functions $f (x) = 1 - e^{- x}$ and $g (x) = - \log (1 - x)$ . Consider the coefficient of $x^{j}$ in the product

\prod_{i = 1}^{m} Q (β_{i} x)

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

If $E$ has the $α_{i}$ as its Chern roots, then the Todd class is:

td (E) = \prod Q (α_{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:

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

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

Properties of the Todd class

The Todd class is multiplicative:

td (E \oplus F) = td (E) \cdot td (F) .

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

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

one obtains

td (T ℂ P^{n}) = {(\frac{ξ}{1 - e^{- ξ}})}^{n + 1} .

Computations of the Todd class

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

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

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

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

where $[H]$ is the hyperplane class in $ℙ^{2}$ restricted to $C$ .

Hirzebruch-Riemann-Roch formula

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

χ (F) = \int_{M} ch (F) \land td (T M),

where $χ (F)$ is its holomorphic Euler characteristic,

χ (F) := \sum_{i = 0}^{{dim}_{ℂ} M} (- 1)^{i} {dim}_{ℂ} H^{i} (M, F),

and $ch (F)$ its Chern character.

Notes

↑ Huybrechts 04, p. 196
↑ Huybrechts 04, Exercise 4.4.5
↑ Huybrechts 04, Exercise 4.4.3
↑ Intersection Theory Class 18, by Ravi Vakil

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, Zbl 0017.18504
Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
M.I. Voitsekhovskii (2001) [1994], "Todd class", Encyclopedia of Mathematics, EMS Press
Huybrechts, Daniel (2004-11-18). Complex geometry: An introduction. Universitext. Springer Science+Business Media. ISBN 978-3540212904.{{cite book}}: CS1 maint: year (link)

[1] Huybrechts 04, p. 196

[:0-2] Huybrechts 04, Exercise 4.4.5

[3] Huybrechts 04, Exercise 4.4.3

[4] Intersection Theory Class 18, by Ravi Vakil