# Derived set (mathematics)

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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 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 Hirzebruch.

## Definition

To define the Todd class 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

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

be the formal power series with the property that the coefficient of xn in Q(x)n+1 is 1 (where the Bi are Bernoulli numbers). Consider the coefficient of xj in the product

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

for any m > j. This is symmetric in the βi and homogeneous of weight j: so can be expressed as a polynomial tdj(p1,...pj) in the elementary symmetric functions p of the β. Then tdj 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: ${\displaystyle Td^{*}(E\oplus F)=Td^{*}(E)\cdot Td^{*}(F)}$

From the Euler exact sequence for the tangent bundle of ${\displaystyle {\mathbb {C}}P^{n}}$

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

and multiplicativity, one obtains

${\displaystyle Td^{*}(T{\mathbb {C}}P^{n})=(\xi /(1-e^{-\xi }))^{n+1},}$

where ${\displaystyle \xi \in H^{2}({\mathbb {C}}P^{n})}$ is the fundamental class of the hyperplane section.[1]

## Hirzebruch-Riemann-Roch formula

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. For any coherent sheaf F on a smooth projective complex manifold M, one has

${\displaystyle \chi (F)=\int _{M}Ch^{*}(F)\wedge Td^{*}(TM),}$
${\displaystyle \chi (F):=\sum _{i=0}^{dim_{\mathbb {C} }M}(-1)^{i}dim_{\mathbb {C} }H^{i}(F),}$

and Ch*(F) its Chern character.

## References

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• F. Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
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