ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 11

structure induces an orientation). Going the other direction, the series Q(x) is

related to the formal group law by

Q(x) =

x

expϕ(x)

,

where expϕ(x) is the inverse to logϕ(x). The second bijection follows in the same

manner, but one needs an even power series to define the stable exponential char-

acteristic class for real vector bundles.

Example 2.9 (K-theory and F×). From (2), the logarithm for the multiplica-

tive formal group law F×(x1,x2) = x1 + x2 − x1x2 is given by

log×(x) =

x

0

dt

1 − t

= − log(1 − x).

Therefore,

exp×(x) = 1 −

e−x,

and the associated power series

Q×(x) =

x

exp×(x)

=

x

1 − e−x

= 1 +

x

2

+

x2

12

−

x4

720

+ · · · ∈ Q x

generates the Todd genus Td. When we evaluate the Todd genus on a Riemann

surface M

2

with genus g,

Td(M

2

) = Q(c1(TM)), [M] = 1 +

1

2

c1(TM) + · · · , [M]

=

1

2

c1(TM), [M] = 1 − g.

In this situation, the Todd genus recovers the standard notion of genus.

Note that even though we started with a Z-valued complex genus, the power

series Q(x) has fractional coeﬃcients. If one is only given Q(x), it is quite surprising

that Todd genus gives integers when evaluated on manifolds with an almost complex

structure on the stable tangent bundle. Another explanation for the integrality is

given by the following important index theorem. In fact, most of the common

genera are equal to the index of some elliptic operator on a manifold (possibly with

G-structure).

Theorem 2.10. (Hirzebruch–Riemann–Roch) Let M be a compact complex

manifold, and let V be a holomorphic vector bundle. Then, the index of the Dol-

beault operator

¯

∂ +

¯∗

∂ on the Dolbeault complex

{Λ0,i

⊗ V }, which equals the Euler

characteristic in sheaf cohomology

H∗(M,

V ), is given by

index(¯

∂ +

¯∗)

∂ = χ(M, V ) = Td(M)ch(V ), [M] ∈ Z.

3. Elliptic genera

Another example of a formal group law comes from the group structure of the

Jacobi quartic elliptic curve. We first start by working over C. Assume δ, ∈ C

and the discriminant Δ =

(δ2

−

)2

= 0. Letting the subscript J stand for Jacobi,

we define

(3) logJ (x) :=

x

0

dt

√

1 − 2δt2 + t4

=

x

0

dt

R(t)

.