• Source: 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 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[H]}{1+d[H]}}\\&=(1+3[H])(1-d[H])\\&=1+(3-d)[H]\end{aligned}}}

where



[
H
]


{\displaystyle [H]}

is the hyperplane class in





P


2




{\displaystyle \mathbb {P} ^{2}}

restricted to



C


{\displaystyle C}

.




Hirzebruch-Riemann-Roch formula



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


Genus of a multiplicative sequence




Notes






References


Todd, J. A. (1937), "The Arithmetical Invariants of Algebraic Loci", Proceedings of the London Mathematical Society, 43 (1): 190–225, doi:10.1112/plms/s2-43.3.190, Zbl 0017.18504
Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)
M.I. Voitsekhovskii (2001) [1994], "Todd class", Encyclopedia of Mathematics, EMS Press

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