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.
To define the Todd class where is a complex vector bundle on a topological space , 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 in is 1, where denotes the -th Bernoulli number. Consider the coefficient of in the product
for any . This is symmetric in the s and homogeneous of weight : so can be expressed as a polynomial in the elementary symmetric functions of the s. Then defines the Todd polynomials: they form a multiplicative sequence with as characteristic power series.
If has the as its Chern roots, then the Todd class
which is to be computed in the cohomology ring of (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:
where the cohomology classes are the Chern classes of , and lie in the cohomology group . If is finite-dimensional then most terms vanish and is a polynomial in the Chern classes.
For any coherent sheaf F on a smooth
compact complex manifold M, one has
where is its holomorphic Euler characteristic,
and its Chern character.
- Todd, J. A. (1937), "The Arithmetical Invariants of Algebraic Loci", Proceedings of the London Mathematical Society, 43 (1): 190–225, doi:10.1112/plms/s2-43.3.190, Zbl 0017.18504
- Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
- M.I. Voitsekhovskii (2001) [1994], "Todd class", Encyclopedia of Mathematics, EMS Press