Todd class

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

Q(x)={\frac {x}{1-e^{-x}}}=\sum _{i=0}^{\infty }{\frac {(-1)^{i}B_{i}}{i!}}x^{i}=1+{\dfrac {x}{2}}+{\dfrac {x^{2}}{12}}-{\dfrac {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 B_i are Bernoulli numbers). Consider the coefficient of x^j in the product

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

for any m > j. This is symmetric in the β_i and homogeneous of weight j: so can be expressed as a polynomial td_j(p₁,...p_j) in the elementary symmetric functions p of the β. Then td_j defines the Todd polynomials: they form a multiplicative sequence with Q as characteristic power series.

If E has the α_i 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 + 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.

Properties of Todd class

The Todd class is multiplicative:

Td^{*}(E\oplus F)=Td^{*}(E)\cdot Td^{*}(F).

Let $\xi \in H^{2}({\mathbb {C}}P^{n})$ be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of ${\mathbb {C}}P^{n}$

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

one obtains ^[1]

Td^{*}(T{\mathbb {C}}P^{n})=\left({\dfrac {\xi }{1-e^{-\xi }}}\right)^{n+1}.

Hirzebruch-Riemann-Roch formula

Main article: Hirzebruch–Riemann–Roch theorem

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}^{{\text{dim}}_{\mathbb {C} }M}(-1)^{i}{\text{dim}}_{\mathbb {C} }H^{i}(F),

and Ch^*(F) its Chern character.

Notes

↑ INTERSECTION THEORY CLASS 18, by Ravi Vakil

References

Todd, J. A. (1937), "The Arithmetical Invariants of Algebraic Loci", Proc. London Math. Soc., 43 (1): 190–225, doi:10.1112/plms/s2-43.3.190, Zbl 0017.18504
F. Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
M.I. Voitsekhovskii (2001), "Todd class", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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