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
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
1 − t
= − log(1 − x).
exp×(x) = 1 −
and the associated power series
1 − e−x
= 1 +
+ · · · ∈ Q x
generates the Todd genus Td. When we evaluate the Todd genus on a Riemann
with genus g,
) = Q(c1(TM)), [M] = 1 +
c1(TM) + · · · , [M]
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
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-
∂ on the Dolbeault complex
⊗ V }, which equals the Euler
characteristic in sheaf cohomology
V ), is given by
∂ = 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 =
= 0. Letting the subscript J stand for Jacobi,
(3) logJ (x) :=
1 − 2δt2 + t4