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 <math>\operatorname{td}(E)</math> where <math>E</math> is a complex vector bundle on a topological space <math>X</math>, 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

<math> 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</math>

be the formal power series with the property that the coefficient of <math>x^n</math> in <math>Q(x)^{n+1}</math> is 1, where <math>B_i</math> denotes the <math>i</math>-th Bernoulli number (with <math>B_1 = +\frac{1}{2}</math>). Consider the coefficient of <math>x^j</math> in the product

<math> \prod_{i=1}^m Q(\beta_i x) \ </math>

for any <math>m > j</math>. This is symmetric in the <math>\beta_i</math>s and homogeneous of weight <math>j</math>: so can be expressed as a polynomial <math>\operatorname{td}_j(p_1,\ldots, p_j)</math> in the elementary symmetric functions <math>p</math> of the <math>\beta_i</math>s. Then <math>\operatorname{td}_j</math> defines the Todd polynomials: they form a multiplicative sequence with <math>Q</math> as characteristic power series.

If <math>E</math> has the <math>\alpha_i</math> as its Chern roots, then the Todd class

<math>\operatorname{td}(E) = \prod Q(\alpha_i)</math>

which is to be computed in the cohomology ring of <math>X</math> (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:

<math>\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 </math>

where the cohomology classes <math>c_i</math> are the Chern classes of <math>E</math>, and lie in the cohomology group <math>H^{2i}(X)</math>. If <math>X</math> is finite-dimensional then most terms vanish and <math>\operatorname{td}(E)</math> is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

<math>\operatorname{td}(E\oplus F) = \operatorname{td}(E)\cdot \operatorname{td}(F).</math>

Let <math>\xi \in H^2({\mathbb C} P^n)</math> be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of <math> {\mathbb C} P^n</math>

<math> 0 \to {\mathcal O} \to {\mathcal O}(1)^{n+1} \to T {\mathbb C} P^n \to 0,</math>

one obtains <ref>Intersection Theory Class 18, by Ravi Vakil</ref>

<math> \operatorname{td}(T {\mathbb C}P^n) = \left( \dfrac{\xi}{1-e^{-\xi}} \right)^{n+1}.</math>

Computations of the Todd class

For any algebraic curve <math>C</math> the Todd class is just <math>\operatorname{td}(C) = 1 + \frac{1}{2} c_1(T_C)</math>. Since <math>C</math> is projective, it can be embedded into some <math>\mathbb{P}^n</math> and we can find <math>c_1(T_C)</math> using the normal sequence

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

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

<math>\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}</math>

where <math>[H]</math> is the hyperplane class in <math>\mathbb{P}^2</math> restricted to <math>C</math>.

Hirzebruch-Riemann-Roch formula

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} For any coherent sheaf F on a smooth compact complex manifold M, one has

<math>\chi(F)=\int_M \operatorname{ch}(F) \wedge \operatorname{td}(TM),</math>

where <math>\chi(F)</math> is its holomorphic Euler characteristic,

<math>\chi(F):= \sum_{i=0}^{\text{dim}_{\mathbb{C}} M} (-1)^i \text{dim}_{\mathbb{C}} H^i(M,F),</math>

and <math>\operatorname{ch}(F)</math> its Chern character.

See also

• Genus of a multiplicative sequence

Notes

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References

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• Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
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