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.

Contents 1 History 2 Definition 3 Properties of Todd class 4 Hirzebruch-Riemann-Roch formula 5 Notes 6 References

[edit] 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.

[edit] 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

Q(x) = x/(1 − e^−x)

considered as a formal power series; the expansion can be made explicit in terms of Bernoulli numbers:

If E has the α_i as its Chern roots, then

td(E) = Π Q(α_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 + 3c₂² − c₄)/720 + ...

where the cohomology classes c_i 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.

[edit] Properties of Todd class

The Todd class is multiplicative:

From the Euler exact sequence for the tangent bundle of

and multiplicativity, one obtains

where is the fundamental class of the hyperplane section.^[1]

[edit] Hirzebruch-Riemann-Roch formula

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

where χ(F) is its holomorphic Euler characteristic,

and Ch^*(F) its Chern character.

[edit] Notes

1. ^ INTERSECTION THEORY CLASS 18, by Ravi Vakil

[edit] References

• J. Todd, The arithmetical theory of algebraic loci, Proc. London Math. Soc., 43 (1937) pp. 190–225
• F. Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
• M.I. Voitsekhovskii (2001), "Todd class", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/T/t092930.htm