Todd class

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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[edit]

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[edit]

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

 Q(x) = \frac{x}{1 - e^{-x}}=\sum_{i=0}^\infty \frac{(-1)^iB_{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

 \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

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[edit]

The Todd class is multiplicative: Td^*(E\oplus F) = Td^*(E)\cdot Td^*(F)

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

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

and multiplicativity, one obtains

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

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

Hirzebruch-Riemann-Roch formula[edit]

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

\chi(F)=\int_M Ch^*(F) \wedge Td^*(TM),

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

\chi(F):= \sum_{i=0}^{dim_{\mathbb{C}}  M} (-1)^i dim_{\mathbb{C}} H^i(F),

and Ch*(F) its Chern character.

See also[edit]

Genus of a multiplicative sequence

Notes[edit]

  1. ^ INTERSECTION THEORY CLASS 18, by Ravi Vakil

References[edit]