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.
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.
The Todd class is multiplicative:
Let be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of
For any algebraic curve the Todd class is just . Since is projective, it can be embedded into some and we can find using the normal sequence
and properties of chern classes. For example, if we have a degree plane curve in , we find the total chern class is
where is the hyperplane class in restricted to .
For any coherent sheaf F on a smooth compact complex manifold M, one has
where is its holomorphic Euler characteristic,
and its Chern character.
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