# 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 ${\displaystyle \operatorname {td} (E)}$ 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

${\displaystyle Q(x)={\frac {x}{1-e^{-x}}}=\sum _{i=0}^{\infty }{\frac {(-1)^{i}B_{i}}{i!}}x^{i}=1+{\dfrac {x}{2}}+{\dfrac {x^{2}}{12}}-{\dfrac {x^{4}}{720}}+\cdots }$

be the formal power series with the property that the coefficient of ${\displaystyle x^{n}}$ in ${\displaystyle Q(x)^{n+1}}$ is 1, where ${\displaystyle B_{i}}$ denotes the i-th Bernoulli number. Consider the coefficient of ${\displaystyle x^{j}}$ in the product

${\displaystyle \prod _{i=1}^{m}Q(\beta _{i}x)\ }$

for any ${\displaystyle m>j}$. This is symmetric in the ${\displaystyle \beta _{i}}$s and homogeneous of weight j: so can be expressed as a polynomial ${\displaystyle \operatorname {td} _{j}(p_{1},\ldots ,p_{j})}$ in the elementary symmetric functions p of the ${\displaystyle \beta _{i}}$s. Then ${\displaystyle \operatorname {td} _{j}}$ defines the Todd polynomials: they form a multiplicative sequence with Q as characteristic power series.

If E has the αi as its Chern roots, then the Todd class

${\displaystyle td(E)=\prod Q(\alpha _{i})}$

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 + c1/2 + (c12+c2)/12 + c1c2/24 + (−c14 + 4c12c2 + c1c3 + 3c22c4)/720 + ...

where the cohomology classes ci are the Chern classes of E, and lie in the cohomology group H2i(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:

${\displaystyle Td^{*}(E\oplus F)=Td^{*}(E)\cdot Td^{*}(F).}$

Let ${\displaystyle \xi \in H^{2}({\mathbb {C} }P^{n})}$ be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of ${\displaystyle {\mathbb {C} }P^{n}}$

${\displaystyle 0\to {\mathcal {O}}\to {\mathcal {O}}(1)^{n+1}\to T{\mathbb {C} }P^{n}\to 0,}$

one obtains [1]

${\displaystyle Td^{*}(T{\mathbb {C} }P^{n})=\left({\dfrac {\xi }{1-e^{-\xi }}}\right)^{n+1}.}$

## Hirzebruch-Riemann-Roch formula

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

${\displaystyle \chi (F)=\int _{M}Ch^{*}(F)\wedge Td^{*}(TM),}$

where ${\displaystyle \chi (F)}$ is its holomorphic Euler characteristic,

${\displaystyle \chi (F):=\sum _{i=0}^{{\text{dim}}_{\mathbb {C} }M}(-1)^{i}{\text{dim}}_{\mathbb {C} }H^{i}(F),}$

and Ch*(F) its Chern character.