LECTURE 5. MULTI-VALUED PERTURBATIONSessentially the arguments in [31]. One considers the moduli spaceMo,k,A(M,w, J,I') {(u,z1, ... ,zk) E B(A,k) : 81(u) E r(u,z1, ... ,zk),
u*[S^2 ] =A, u is somewhere injective}.
221The group G acts freely on this space and, for a generic perturbation r, the quotient
Mo,k,A(M,w, J,r) = M. 0 ,k,A(M,w, J ,I')/G
is a branched manifold of dimension
dim Mo,k,A(M,w, J , I') = 2n + 2k + 2c1(A) - 6,
with rational weight >. : M 0 k A --+ Q. Note that this space M 0 k A is an open
subset of the space Mo,k,A(M, w, J , I') of all stable (J,I')-holomo~phic curves of
genus zero with k marked points in M which represent the class A. The definitions
and theorems are as in Section 4 for the unperturbed case. In particular, the moduli
space Mo,k,A is a compact metrizable space, it is a union of branched manifolds
corresponding to the different tree structures, and M 0 k A is the "top dimensional
stratum" of Mo,k,A· ' 'Given cohomology classes a 1 , ... , ak E H* ( M, Z), choose submanifolds (or
cycles) Nj c M in general position which are Poincare dual to the aj, i.e. aj =
PD([Nj]). If
k
L deg(aj ) = 2n + 2k + 2c 1 (A) - 6,
j=l
then the subset of all tuples [u, z1, ... , zk] E Mo,k,A with u( zj) E Nj is a compact
zero dimensional branched manifold with a natural orientations. Thus M 0 k A is a
finite set. But in the zero dimensional case a single isolated point may b~l~ng to
several branches and each of the branches may carry different orientations, which
in the zero dimensional case means different signs. Geometrically, this means that
several branches of the multi-valued perturbation may have a common zero but not
have the same differential at this zero. Given a solution [u, z1, ... , zk] E Mo,k,A
with u( Zj) E Nj, let II,... , Im denote the local branches of the perturbation r near
(u, z 1 , ... , zk), and let >. 1 , ... , Am be the corresponding weights. For each branch
I i which satisfies li(u, z 1 , ... , zk) = 81(u) there is a sign c i E {±1}, determined by
the orientation of this branch, and we define
i E{l, ... cm}
8J(u)=1;(u,z 1 , ... ,zk)Counting the solutions with these rational weights gives rise to the rational
Gromov-Witten invariant
(54)
[u,z 1 , ... ,zkJEMo, k , A
u(z1 )ENj
One can prove, proceeding as in [31] and using Lemma 5. 11 , that the right hand
side is i~dependent of the cycles N j , the almost complex structure J, and the
perturbation I', used to define it. Details will be carried out in [22].
Remark 5.12. A crucial point in the construction of the Gromov-Witten invari-
ants is the fact that, generically, there are only finitely many equivalence classes of
solutions ( u , z 1 , ... , zk) of (50) which satisfy u(zi ) E Ni , and that u is somewhere