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 ${\displaystyle E}$ is a complex vector bundle on a topological space ${\displaystyle X}$, 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

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

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 ${\displaystyle i}$-th Bernoulli number (with ${\displaystyle B_{1}=+{\frac {1}{2}}}$). 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 ${\displaystyle j}$: so can be expressed as a polynomial ${\displaystyle \operatorname {td} _{j}(p_{1},\ldots ,p_{j})}$ in the elementary symmetric functions ${\displaystyle p}$ of the ${\displaystyle \beta _{i}}$s. Then ${\displaystyle \operatorname {td} _{j}}$ defines the Todd polynomials: they form a multiplicative sequence with ${\displaystyle Q}$ as characteristic power series.

If ${\displaystyle E}$ has the ${\displaystyle \alpha _{i}}$ as its Chern roots, then the Todd class

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

which is to be computed in the cohomology ring of ${\displaystyle X}$ (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:

${\displaystyle \operatorname {td} (E)=1+{\frac {c_{1}}{2}}+{\frac {c_{1}^{2}+c_{2}}{12}}+{\frac {c_{1}c_{2}}{24}}+{\frac {-c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}}{720}}+\cdots }$

where the cohomology classes ${\displaystyle c_{i}}$ are the Chern classes of ${\displaystyle E}$, and lie in the cohomology group ${\displaystyle H^{2i}(X)}$. If ${\displaystyle X}$ is finite-dimensional then most terms vanish and ${\displaystyle \operatorname {td} (E)}$ is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

${\displaystyle \operatorname {td} (E\oplus F)=\operatorname {td} (E)\cdot \operatorname {td} (F).}$

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]

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

Computations of the Todd class

For any algebraic curve ${\displaystyle C}$ the Todd class is just ${\displaystyle \operatorname {td} (C)=1+{\frac {1}{2}}c_{1}(T_{C})}$. Since ${\displaystyle C}$ is projective, it can be embedded into some ${\displaystyle \mathbb {P} ^{n}}$ and we can find ${\displaystyle c_{1}(T_{C})}$ using the normal sequence

${\displaystyle 0\to T_{C}\to T_{\mathbb {P^{n}} }|_{C}\to N_{C/\mathbb {P} ^{n}}\to 0}$

and properties of chern classes. For example, if we have a degree ${\displaystyle d}$ plane curve in ${\displaystyle \mathbb {P} ^{2}}$, we find the total chern class is

${\displaystyle {\begin{aligned}c(T_{C})&={\frac {c(T_{\mathbb {P} ^{2}}|_{C})}{c(N_{C/\mathbb {P} ^{2}})}}\\&={\frac {1+3[H]}{1+d[H]}}\\&=(1+3[H])(1-d[H])\\&=1+(3-d)[H]\end{aligned}}}$

where ${\displaystyle [H]}$ is the hyperplane class in ${\displaystyle \mathbb {P} ^{2}}$ restricted to ${\displaystyle C}$.

Hirzebruch-Riemann-Roch formula

Main article: Hirzebruch–Riemann–Roch theorem

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

${\displaystyle \chi (F)=\int _{M}\operatorname {ch} (F)\wedge \operatorname {td} (TM),}$

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

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

and ${\displaystyle \operatorname {ch} (F)}$ its Chern character.

See also

• Genus of a multiplicative sequence