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 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 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 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 α -->
i
{\displaystyle \alpha _{i}}
as its Chern roots , then the Todd class
td
-->
(
E
)
=
∏ ∏ -->
Q
(
α 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 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
[ 1]
td
-->
(
T
C
P
n
)
=
(
ξ ξ -->
1
− − -->
e
− − -->
ξ 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}
.
For any coherent sheaf F on a smooth
compact complex manifold M , one has
χ χ -->
(
F
)
=
∫ ∫ -->
M
ch
-->
(
F
)
∧ ∧ -->
td
-->
(
T
M
)
,
{\displaystyle \chi (F)=\int _{M}\operatorname {ch} (F)\wedge \operatorname {td} (TM),}
where
χ χ -->
(
F
)
{\displaystyle \chi (F)}
is its holomorphic Euler characteristic ,
χ 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
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