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 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 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, let

$Q(x) = \frac{x}{1 - e^{-x}}=\sum_{i=0}^\infty \frac{(-1)^iB_{i}}{i!}x^{i} = 1 +\dfrac{x}{2}+\dfrac{x^2}{12}-\dfrac{x^4}{720}+\cdots$

be the formal power series with the property that the coefficient of xⁿ in Q(x)ⁿ⁺¹ is 1 (where the B_i are Bernoulli numbers). 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 β_i and homogeneous of weight j: so can be expressed as a polynomial td_j(p₁,...p_j) in the elementary symmetric functions p of the β. 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

$td(E) = \prod Q(\alpha_i)$

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:

td(E) = 1 + c₁/2 + (c₁²+c₂)/12 + c₁c₂/24 + (−c₁⁴ + 4c₁²c₂ + c₁c₃ + 3c₂² − c₄)/720 + ...

where the cohomology classes c_i are the Chern classes of E, and lie in the cohomology group H²ⁱ(X). If X is finite-dimensional then most terms vanish and td(E) is a polynomial in the Chern classes.

Properties of Todd class

The Todd class is multiplicative:

$Td^*(E\oplus F) = Td^*(E)\cdot Td^*(F).$

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

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

one obtains

$Td^*(T {\Bbb C}P^n) = \left( \dfrac{\xi}{1-e^{-\xi}} \right)^{n+1}.$

Hirzebruch-Riemann-Roch formula

Main article: Hirzebruch–Riemann–Roch theorem

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

$\chi(F)=\int_M Ch^*(F) \wedge Td^*(TM),$

where $\chi(F)$ is its holomorphic Euler characteristic,

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

and Ch^*(F) its Chern character.

See also

Genus of a multiplicative sequence

References

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