# Derived set (mathematics)

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

be the formal power series with the property that the coefficient of *x*^{n} in *Q*(*x*)^{n+1} is 1 (where the *B*_{i} are Bernoulli numbers). Consider the coefficient of *x*^{j} in the product

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}(*p*_{1},...*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**

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 +*c*_{1}/2 + (*c*_{1}^{2}+*c*_{2})/12 +*c*_{1}*c*_{2}/24 + (−*c*_{1}^{4}+ 4*c*_{1}^{2}*c*_{2}+*c*_{1}*c*_{3}+ 3*c*_{2}^{2}−*c*_{4})/720 + ...

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 *td*(*E*) is a polynomial in the Chern classes.

## Properties of Todd class

The Todd class is multiplicative:

From the Euler exact sequence for the tangent bundle of

and multiplicativity, one obtains

where is the fundamental class of the hyperplane section.^{[1]}

## Hirzebruch-Riemann-Roch formula

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

where is its holomorphic Euler characteristic,

and *Ch ^{*}(F)* its Chern character.

## See also

Genus of a multiplicative sequence

## Notes

## References

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