Todd class

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Characteristic class in algebraic topology

title: "Todd class" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["characteristic-classes"] description: "Characteristic class in algebraic topology" topic_path: "general/characteristic-classes" source: "https://en.wikipedia.org/wiki/Todd_class" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 \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}}=\sum_{i=0}^\infty \frac{B_i}{i!}x^i = 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 (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_is 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_is. 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 + \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 sequence0 \to T_C \to T_\mathbb{P^n}|C \to N{C/\mathbb{P}^n} \to 0and 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

Main article: Hirzebruch–Riemann–Roch theorem

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.

Notes

References

References

  1. Huybrechts 04, p. 196
  2. Huybrechts 04, Excercise 4.4.5
  3. Huybrechts 04, Excercise 4.4.3
  4. [http://math.stanford.edu/~vakil/245/245class18.pdf Intersection Theory Class 18], by [[Ravi Vakil]]

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