Todd class

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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]\displaystyle{ \operatorname{td}(E) }[/math] where [math]\displaystyle{ E }[/math] is a complex vector bundle on a topological space [math]\displaystyle{ 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]\displaystyle{ Q(x) = \frac{x}{1 - e^{-x}}=1+\dfrac{x}{2}+\sum_{i=1}^\infty \frac{B_{2i}}{(2i)!}x^{2i} = 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]\displaystyle{ x^n }[/math] in [math]\displaystyle{ Q(x)^{n+1} }[/math] is 1, where [math]\displaystyle{ B_i }[/math] denotes the [math]\displaystyle{ i }[/math]-th Bernoulli number. Consider the coefficient of [math]\displaystyle{ x^j }[/math] in the product

[math]\displaystyle{ \prod_{i=1}^m Q(\beta_i x) \ }[/math]

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

If [math]\displaystyle{ E }[/math] has the [math]\displaystyle{ \alpha_i }[/math] as its Chern roots, then the Todd class

[math]\displaystyle{ \operatorname{td}(E) = \prod Q(\alpha_i) }[/math]

which is to be computed in the cohomology ring of [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ c_i }[/math] are the Chern classes of [math]\displaystyle{ E }[/math], and lie in the cohomology group [math]\displaystyle{ H^{2i}(X) }[/math]. If [math]\displaystyle{ X }[/math] is finite-dimensional then most terms vanish and [math]\displaystyle{ \operatorname{td}(E) }[/math] is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

[math]\displaystyle{ \operatorname{td}(E\oplus F) = \operatorname{td}(E)\cdot \operatorname{td}(F). }[/math]

Let [math]\displaystyle{ \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]\displaystyle{ {\mathbb C} P^n }[/math]

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

one obtains [1]

[math]\displaystyle{ \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]\displaystyle{ C }[/math] the Todd class is just [math]\displaystyle{ \operatorname{td}(C) = 1 + c_1(T_C) }[/math]. Since [math]\displaystyle{ C }[/math] is projective, it can be embedded into some [math]\displaystyle{ \mathbb{P}^n }[/math] and we can find [math]\displaystyle{ c_1(T_C) }[/math] using the normal sequence

[math]\displaystyle{ 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]\displaystyle{ d }[/math] plane curve in [math]\displaystyle{ \mathbb{P}^2 }[/math], we find the total chern class is

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

Hirzebruch-Riemann-Roch formula

Main page: Hirzebruch–Riemann–Roch theorem

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

[math]\displaystyle{ \chi(F)=\int_M \operatorname{ch}(F) \wedge \operatorname{td}(TM), }[/math]

where [math]\displaystyle{ \chi(F) }[/math] is its holomorphic Euler characteristic,

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

and [math]\displaystyle{ \operatorname{ch}(F) }[/math] its Chern character.

See also


Notes

  1. ↑ Intersection Theory Class 18, by Ravi Vakil

References



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