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Wikipedia / 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

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-th Bernoulli number in the product

in the elementary symmetric functions defines the Todd polynomials: they form a multiplicative sequence as characteristic power series.

as its Chern roots, then the Todd class

which is to be computed in the cohomology ring (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:

is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

one obtains

Computations of the Todd class

using the normal sequence, we find the total chern class is.

Hirzebruch-Riemann-Roch formula

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

is its holomorphic Euler characteristic,

its Chern character.

See also

Notes

References