LECTURE 5. MULTI-VALUED PERTURBATIONS 207Example 5.1. Suppose that (M,w) is an 8-dimensional symplectic manifold whichsatisfies ( 4) with r < 0 and has minimal Chern number N = l. The simplest such
example is a hypersurface of degree 7 in <CP^5 (see Exercise 1.5). In this case J-holomorphic spheres of Chern number -1 are isolated and, for a generic almost
complex structure J E J(M,w), there will be finitely many such spheres which
cannot be destroyed by a perturbation of J. Suppose that v : 82 ___, M is such acurve and that f : 82 ---> 82 is a rational map of degree k ;::: 2. Then v o f is a
J-holomorphic curve of Chern number c 1 (v o !) = -k. Thus the J-holomorphic
curves of Chern number -k, modulo reparametrization, form a moduli space of
dimension dim Ratk - dim G = 4k - 4 while the virtual dimension is 2 - 2k < 0. D
5.2. Multi-valued perturbations
A smooth map v: 82 ---> Mis a J-holomorphic sphere if BJ(v) = 0, where
BJ(v) = ~(dv + J 0 dv 0 i) E n°'^1 (S^2 , v*TM).
From an abstract point of view there is an infinite dimensional vector bundle E ---> B
over the space B = Map(S^2 , M) with fibers Ev = n,o,i (8^2 , v*T M), the nonlinear
operator BJ : B---> E is a section of this bundle, and the J-holomorphic spheres are
the zeros of this section. This section has the following crucial properties.
(i) BJ is a Fredholm section. This means that the vertical differentialDv = DBJ(v): TvB = C^00 (S^2 ,vTM)--+ Ev= D.^0 '^1 (8^2 ,vTM)
is a Fredholm operator between appropriate Sobolev completions. Its Fred-holm index is given by index Dv = 2n + 2c 1 ( v), where 2n = dim M.
(ii) For a generic almost complex structure J the restriction of BJ to the subset
B^8 of simple maps is transverse to the zero section. This means that the
vertical differential D v is surjective for every simple J-holomorphic sphere
v.(iii) There is an action of the group G = PSL(2, <C) of Mi:ibius transformations
on both Band E. The section BJ is equivariant under this action, i.e.BJ(v o <p) = <p*BJ(v)
for v E B and <p E G.As noted in the previous section, the transversality statement does not extend to
all of B. Hence a different perturbation/: B---> E must be found to make BJ - I
transverse to the zero section over all of B. This perturbation must satisfy all three
requirements of transversality, equivariance, and of being "lower order", so that the
perturbed equation is still Fredholm with the same index. Unfortunately, all three
requirements cannot be fulfilled, in general, by single-valued perturbations. This
phenomenon is already apparent in the finite dimensional case where transversality
cannot, in general, be achieved while preserving equivariance. It will, however, be
possible. to achieve transversality by choosing an equivariant multi-valued pertur-
bation
r: B ___, 2£.
Thus, for each v E B, r( v) is a finite subset of Ev = n,o,^1 ( 82 , v*T M). The perturbed
Cauchy-Riemann equatl.ons take the form
(47)