Todd class explained

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

\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 Q(x) = (x)/(1- e ^{-x}) =1+ \dfrac{x}{ 2 } + \sum ^{inf ty} _{i =1} (B _{2i})/(( 2 i)!) x ^{2i} = 1 + \dfrac{x}{ 2 } + \dfrac{x ^{2}{12}-\dfrac{x} ^{4}{720}+ …} 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. Consider the coefficient of x ^{j} in the product \prod ^{m} _{i =1} Q(\beta _{i} x)

for any

m > j . This is symmetric in the \beta _{i} s and homogeneous of weight j : so can be expressed as a polynomial \operatorname{td} _{j(p} _{1,\ldots,} p _{j)} in the elementary symmetric functions p of the \beta _{i} s. Then \operatorname{td} _{j} defines the Todd polynomials: they form a multiplicative sequence with Q as characteristic power series. If E has the \alpha _{i} as its Chern roots, then the Todd class\operatorname{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:

\operatorname{td}(E) = 1 + (c _{1})/(2) + (c ^{2} _{1} + c _{2})/(12) + (c _{1c} _{2})/(24) + (- c ^{4} _{1} + 4 c ^{2} _{1} c _{2} + c _{1c} _{3} + 3 c ^{2} _{2} - c _{4})/(720) + …

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 \operatorname{td}(E) is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

\operatorname{td}(E ⊕ F) = \operatorname{td}(E) ⋅ \operatorname{td}(F).

Let

\xi \in H ^{2({C}P} ^{n)} be the fundamental class of the hyperplane section.From multiplicativity and the Euler exact sequence for the tangent bundle of { C} P ^{n} 0 \to { l O} \to { l O}( 1 ) ^{n+1} \to T { C} P ^{n} \to 0 , one obtains^[1] \operatorname{td}(T { C}P ^{n)} = \left( \dfrac{\xi}{ 1- e ^{-\xi} } \right)^.

Computations of the Todd class

For any algebraic curve

C the Todd class is just \operatorname{td}(C) = 1 + c _{1(T} _{C)} . Since C is projective, it can be embedded into some P ^{n} and we can find c _{1(T} _{C)} using the normal sequence

0 \to T _{C} \to T ^{n|} _{C} \to N _{C/ P ^{n}} \to 0

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

\begin{align} c(T _{C)} & = (c(T _{P ^{2}} | _{C)})/(c(N _{C/ P ^{2}} )) \\ & = (1+3 [H])/(1+ d[H]) \\ & = ( 1+3 [H])( 1- d[H]) \\ & = 1 + ( 3- d)[H] \end{align}

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

Hirzebruch-Riemann-Roch formula

See main article: Hirzebruch–Riemann–Roch theorem. For any coherent sheaf F on a smooth compact complex manifold M, one has

\chi(F) = \int _{M} \operatorname{ch}(F) \wedge \operatorname{td}(TM), where \chi(F) is its holomorphic Euler characteristic,\chi(F): = \sum ^{dim _{C} M} _{i =0} ( -1 ) ^{i} dim _{C} H ^{i(M,F),} and \operatorname{ch}(F) its Chern character.

See also

• Genus of a multiplicative sequence

Notes

1. http://math.stanford.edu/~vakil/245/245class18.pdf Intersection Theory Class 18

References

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