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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 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 td ( E ) {\displaystyle \operatorname {td} (E)} where E {\displaystyle E} is a complex vector bundle on a topological space X {\displaystyle 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 + x 2 + i = 1 B 2 i ( 2 i ) ! x 2 i = 1 + x 2 + x 2 12 x 4 720 + {\displaystyle Q(x)={\frac {x}{1-e^{-x}}}=1+{\dfrac {x}{2}}+\sum _{i=1}^{\infty }{\frac {B_{2i}}{(2i)!}}x^{2i}=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 x n {\displaystyle x^{n}} in Q ( x ) n + 1 {\displaystyle Q(x)^{n+1}} is 1, where B i {\displaystyle B_{i}} denotes the i {\displaystyle i} -th Bernoulli number. Consider the coefficient of x j {\displaystyle x^{j}} in the product

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

for any m > j {\displaystyle m>j} . This is symmetric in the β i {\displaystyle \beta _{i}} s and homogeneous of weight j {\displaystyle j} : so can be expressed as a polynomial td j ( p 1 , , p j ) {\displaystyle \operatorname {td} _{j}(p_{1},\ldots ,p_{j})} in the elementary symmetric functions p {\displaystyle p} of the β i {\displaystyle \beta _{i}} s. Then td j {\displaystyle \operatorname {td} _{j}} defines the Todd polynomials: they form a multiplicative sequence with Q {\displaystyle Q} as characteristic power series.

If E {\displaystyle E} has the α i {\displaystyle \alpha _{i}} as its Chern roots, then the Todd class

td ( E ) = Q ( α i ) {\displaystyle \operatorname {td} (E)=\prod Q(\alpha _{i})}

which is to be computed in the cohomology ring of X {\displaystyle 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 + c 1 2 + c 1 2 + c 2 12 + c 1 c 2 24 + c 1 4 + 4 c 1 2 c 2 + c 1 c 3 + 3 c 2 2 c 4 720 + {\displaystyle \operatorname {td} (E)=1+{\frac {c_{1}}{2}}+{\frac {c_{1}^{2}+c_{2}}{12}}+{\frac {c_{1}c_{2}}{24}}+{\frac {-c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}}{720}}+\cdots }

where the cohomology classes c i {\displaystyle c_{i}} are the Chern classes of E {\displaystyle E} , and lie in the cohomology group H 2 i ( X ) {\displaystyle H^{2i}(X)} . If X {\displaystyle X} is finite-dimensional then most terms vanish and td ( E ) {\displaystyle \operatorname {td} (E)} is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative:

td ( E F ) = td ( E ) td ( F ) . {\displaystyle \operatorname {td} (E\oplus F)=\operatorname {td} (E)\cdot \operatorname {td} (F).}

Let ξ H 2 ( C P n ) {\displaystyle \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 C P n {\displaystyle {\mathbb {C} }P^{n}}

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

one obtains

td ( T C P n ) = ( ξ 1 e ξ ) n + 1 . {\displaystyle \operatorname {td} (T{\mathbb {C} }P^{n})=\left({\dfrac {\xi }{1-e^{-\xi }}}\right)^{n+1}.}

Computations of the Todd class

For any algebraic curve C {\displaystyle C} the Todd class is just td ( C ) = 1 + 1 2 c 1 ( T C ) {\displaystyle \operatorname {td} (C)=1+{\frac {1}{2}}c_{1}(T_{C})} . Since C {\displaystyle C} is projective, it can be embedded into some P n {\displaystyle \mathbb {P} ^{n}} and we can find c 1 ( T C ) {\displaystyle c_{1}(T_{C})} using the normal sequence

0 T C T P n | C N C / P n 0 {\displaystyle 0\to T_{C}\to T_{\mathbb {P^{n}} }|_{C}\to N_{C/\mathbb {P} ^{n}}\to 0}

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

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 ] {\displaystyle {\begin{aligned}c(T_{C})&={\frac {c(T_{\mathbb {P} ^{2}}|_{C})}{c(N_{C/\mathbb {P} ^{2}})}}\\&={\frac {1+3}{1+d}}\\&=(1+3)(1-d)\\&=1+(3-d)\end{aligned}}}

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

Hirzebruch-Riemann-Roch formula

Main article: Hirzebruch–Riemann–Roch theorem

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

χ ( F ) = M ch ( F ) td ( T M ) , {\displaystyle \chi (F)=\int _{M}\operatorname {ch} (F)\wedge \operatorname {td} (TM),}

where χ ( F ) {\displaystyle \chi (F)} is its holomorphic Euler characteristic,

χ ( F ) := i = 0 dim C M ( 1 ) i dim C H i ( M , F ) , {\displaystyle \chi (F):=\sum _{i=0}^{{\text{dim}}_{\mathbb {C} }M}(-1)^{i}{\text{dim}}_{\mathbb {C} }H^{i}(M,F),}

and ch ( F ) {\displaystyle \operatorname {ch} (F)} its Chern character.

See also

Notes

  1. Intersection Theory Class 18, by Ravi Vakil

References

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