LECTURE 5. MULTI-VALUED PERTURBATIONS 209• The vertical differential D1i(v): TvBk,p--> £i-l,p is a compact oper-
ator for v E Uk,p and i = 1, ... , m.^2
(Transversality) If v : 82 --> M is a ( J, r) holomorphic curve and 81 ( v) =
/i(v) then the vertical differential Di,v = D81(v) - D/i(v) is surjective.
(Free) If v : 82 --> M is a ( J , I') holomorphic curve and i.p E G such that
v o rp = v then rp = id.
Perhaps the most surprising condition here is the relatively complicated form
of the "local structure" axiom. The reason for this formulation is the fact that
the action of G on Bk,p is not smooth. To be more precise, let v : 82 --> M be a
Wk,P_function and t 1-+ 'Pt be a smooth path in G. Then the function JR --> Bk,p :
t 1-+ Vt = v o 'Pt is only continuous but not differentiable, because differentiatingwith respect to t we obtain a vector field along Vt of class wk-l,p but not in
Wk,P(8^2 ,vt*TM) = Tv,Bk,p. In other words, the function
Bk,p x G _, Bk,p: (v, rp) I-+ v o rp
is only continuous, but the function Bk,p x G --> Bk-£,p : (v, rp) 1-+ v o rp is of classce for 0 ::; f ::; k. This differentiability of the action at the expense of the loss
of differentibility of v is inherited by our perturbation. The compactness of thevertical differential D/i ( v) follows essentially from the fact that the group G is
finite dimensional.
Remark 5.2. Since the space B = Map(8^2 ,M) is separable (with the c^0 -
topology) it can be covered by countably many open sets U(j) which satisfy the
requirements of the "local structure" axiom. Hence there exists a collection of local
sections /i : Ui --> £ and rational numbers Ai > 0, indexed by a countable set I ,
and a decomposition of the index set I= LJj I(j) into finite sets, such that each /i
satisfies the smoothness requirement of the "local structure" axiom, Ui = U(j) for
i E I(j), I.:iEJ(j) Ai = 1, and r( v) = bi ( v) i E I(j)}, A( v, TJ) = I: iEI(j) Ai for
"f;(v)=17
v E U(j) and T/ E Ev. D.
We shall have to deal with three problems: the existence of .perturbations
which satisfy the above axioms, the properties of the resulting moduli spaces, and
the compactness problem for sequences of (J, r)-holomorphic curves. These will be
discussed in the next three sections.
5.3. Local slicesThe goal of this section is to prove the existence of perturbations r which satisfy
the above requirements. Our construction is based on local slices for the G-action
on B. Let us fix a smooth function u: S^2 --> M with finite isotropy subgroup
Gu={rpEG: uorp=u}.
(^2) The vertical differential of a continuously differentiable section"( : 5 k,p --+ £i,p (with j :::; k)
is defined as the linear operator D"((v) : Tv B k,p--+ £l'P defined by
D"((v)~ = !!:_I <Pv(t~)-^1 '"Y(expv(t~))
dt t=O
for~ E Wk,P(S^2 ,vTM). Here <Pv(~) : vTM--+ expv(~)*TM denotes parallel transport with
respect to the Hermitian connection induced by J. If "i7 denotes the Levi-Civita connection of the
metric (-, ·) = w(· , J ·) then the Hermitian connection is defined by "i7^1 = "i7 - ~J"il J.