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

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

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

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:

Let be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of

one obtains [1]

Hirzebruch-Riemann-Roch formula[edit]

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

where is its holomorphic Euler characteristic,

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]