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.
To define the Todd class formula_1 where formula_2 is a complex vector bundle on a topological space formula_3, 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 formula_5 in formula_6 is 1, where formula_7 denotes the formula_8-th Bernoulli number. Consider the coefficient of formula_9 in the product 
for any formula_11. This is symmetric in the formula_12s and homogeneous of weight formula_13: so can be expressed as a polynomial formula_14 in the elementary symmetric functions formula_15 of the formula_12s. Then formula_17 defines the Todd polynomials: they form a multiplicative sequence with formula_18 as characteristic power series.
If formula_2 has the formula_20 as its Chern roots, then the Todd class
which is to be computed in the cohomology ring of formula_3 (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 formula_24 are the Chern classes of formula_2, and lie in the cohomology group formula_26. If formula_3 is finite-dimensional then most terms vanish and formula_1 is a polynomial in the Chern classes.
Properties of Todd class.
The Todd class is multiplicative: 
Let formula_30 be the fundamental class of the hyperplane section.
From multiplicativity and the Euler exact sequence for the tangent bundle of formula_31
one obtains
Computations of Todd Class.
For any algebraic curve formula_34 the Todd class is just formula_35. Since formula_34 is projective, it can be embedded into some formula_37 and we can find formula_38 using the normal sequenceformula_39and properties of chern classes. For example, if we have a degree formula_40 plane curve in formula_41, we find the total chern class isformula_42where formula_43 is the hyperplane class in formula_41 restricted to formula_34.
Hirzebruch-Riemann-Roch formula.
For any coherent sheaf "F" on a smooth 
compact complex manifold "M", one has
where formula_47 is its holomorphic Euler characteristic,
and formula_49 its Chern character.