Todd class

 ( Characteristic Classes ) 

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 , and is encountered where Chern classes exist — most notably in differential topology, the theory of 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.

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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.

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Definition To define the Todd class $\backslash operatorname\{td\}(E)$ 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 , by means of a general device of characteristic class theory, the use of Chern roots (aka, the splitting principle). For the definition, letHuybrechts 04, p. 196

:$Q(x)\; =\; \backslash frac\{x\}\{1\; -\; e^\{-x\}\}=\backslash sum\_\{i=0\}^\backslash infty\; \backslash frac\{B\_i\}\{i!\}x^i\; =\; 1\; +\backslash dfrac\{x\}\{2\}+\backslash dfrac\{x^2\}\{12\}-\backslash dfrac\{x^4\}\{720\}+\backslash cdots$

be the formal power series with the property that the coefficient of $x^n$ in $Q(x)^\{n+1\}$ is 1, where $B\_i$ denotes the $i$-th Bernoulli number (with $B\_1\; =\; +\backslash frac\{1\}\{2\}$). Consider the coefficient of $x^j$ in the product

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

for any $m\; >\; j$. This is symmetric in the $\backslash beta\_i$s and homogeneous of weight $j$: so can be expressed as a polynomial $\backslash operatorname\{td\}\_j(p\_1,\backslash ldots,\; p\_j)$ in the elementary symmetric functions $p$ of the $\backslash beta\_i$s. Then $\backslash operatorname\{td\}\_j$ defines the Todd polynomials: they form a multiplicative sequence with $Q$ as characteristic power series.

If $E$ has the $\backslash alpha\_i$ as its Chern roots, then the Todd class

$\backslash operatorname\{td\}(E)\; =\; \backslash prod\; Q(\backslash alpha\_i)$

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:Huybrechts 04, Excercise 4.4.5

$\backslash operatorname\{td\}(E)\; =\; 1\; +\; \backslash frac\{c\_1\}\{2\}\; +\; \backslash frac\{c\_1^2\; +c\_2\}\{12\}\; +\; \backslash frac\{c\_1c\_2\}\{24\}\; +\; \backslash frac\{-c\_1^4\; +\; 4\; c\_1^2\; c\_2\; +\; c\_1c\_3\; +\; 3c\_2^2\; -\; c\_4\}\{720\}\; +\; \backslash cdots$

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

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Properties of the Todd class The Todd class is multiplicative:Huybrechts 04, Excercise 4.4.3

:$\backslash operatorname\{td\}(E\backslash oplus\; F)\; =\; \backslash operatorname\{td\}(E)\backslash cdot\; \backslash operatorname\{td\}(F).$

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

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

one obtains Intersection Theory Class 18, by Ravi Vakil

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

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Computations of the Todd class For any algebraic curve $C$ the Todd class is just $\backslash operatorname\{td\}(C)\; =\; 1\; +\; \backslash frac\{1\}\{2\}\; c\_1(T\_C)$. Since $C$ is projective, it can be embedded into some $\backslash mathbb\{P\}^n$ and we can find $c\_1(T\_C)$ using the normal sequence

$0\; \backslash to\; T\_C\; \backslash to\; T\_\backslash mathbb\{P^n\}|\_C\; \backslash to\; N\_\{C/\backslash mathbb\{P\}^n\}\; \backslash to\; 0$

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

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

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Hirzebruch-Riemann-Roch formula For any coherent sheaf F on a smooth compact complex manifold M, one has

:$\backslash chi(F)=\backslash int\_M\; \backslash operatorname\{ch\}(F)\; \backslash wedge\; \backslash operatorname\{td\}(TM),$

where $\backslash chi(F)$ is its holomorphic Euler characteristic,

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

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

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See also

• Genus of a multiplicative sequence

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Notes

• Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
• (2004). 9783540212904, Springer Science+Business Media. . ISBN 9783540212904

Categories: Characteristic Classes

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