Enumerative vs. Symplectic Invariants and Obstruction Bundles

77 Pages·0.75 MB·English

by  A. Zinger

Enumerative vs. Symplectic Invariants and Obstruction Bundles
Aleksey Zinger ∗
2
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February 1, 2008
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a
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6 Contents
2

] 1 Introduction 2
G
1.1 Background and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
S
1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
.
h 1.3 Fundamental Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
t
a
m 2 Spaces of Bubble Maps 6
[ 2.1 Bubble Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Basic Gluing Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1
v 2.3 Curves with Marked Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 2.4 Bubble Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
2.5 Stratums of Bubble Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2
1
0 3 The Gluing Construction and the Obstruction Bundle 17
2
3.1 Summary and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
0
/ 3.2 The Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
h
3.3 Construction of Nearly Holomorphic Bubble Maps . . . . . . . . . . . . . . . . . . . 24
t
a
3.4 Scale of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
m
3.5 Obstruction Bundle Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
:
v 3.6 The Gluing Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
i
X 3.7 An Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
r 3.8 The Orientation of Σ,tν,λ∗(µ) and the Gluing Map . . . . . . . . . . . . . . . . . . 38
a M
3.9 Gluing Maps for Spaces ¯(0)(µ) and Orientations . . . . . . . . . . . . . . . . . . . . 46
UT
4 Technical Issues 49
4.1 Continuity of the Gluing Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Injectivity of the Gluing Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 The Basic Gluing Map and the Space of Balanced Maps . . . . . . . . . . . . . . . . 56
4.4 Gromov Convergence and the Lp-norm of the Differential . . . . . . . . . . . . . . . 60
4.5 Surjectivity of the Gluing Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
∗Partially supported byNSFGraduate Research Fellowship and NSFgrant DMS-9803166
5 Appendix 67
5.1 Properties of Smooth Families of Metrics on Σ . . . . . . . . . . . . . . . . . . . . . 67
5.2 Sobolev Inequalities for the Metrics g . . . . . . . . . . . . . . . . . . . . . . . . . . 71
υ
5.3 Elliptic Estimates for the Metrics g . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
υ
5.4 Fiber-Uniform Inverse for the Operator D . . . . . . . . . . . . . . . . . . . . . . . 75
υ
1 Introduction
1.1 Background and Results
Suppose (Σ,j) is a nonsingular Riemann surface of genus g 2 and (V,J,ω) is a Kahler manifold
≥
of complex dimension n. If λ H (V;Z), denote by (V) the set of simple (J,j)-holomorphic
2 Σ,λ
∈ H
maps u from Σ to V such that u [Σ]= λ. Let µ = (µ ,... ,µ ) be an N-tuple of proper oriented
∗ 1 N
submanifolds of V such that
l=N
codim µ codim µ = 2 c (V,J),λ n(g 1)+N . (1.1)
l 1
≡ h i− −
l=1
X (cid:0) (cid:1)
For many Kahler manifolds (V,J,ω) and choice of constraints µ, the cardinality of the set
(µ)= (Σ;y ,... ,y ;u): u (V); y Σ, u(y ) µ l = 1,... ,N (1.2)
Σ,λ 1 N Σ,λ l l l
H ∈H ∈ ∈ ∀
(cid:8) (cid:9)
is finite and depends only on the homology classes of µ ,... ,µ . The cardinality (µ) of the
1 N Σ,λ
|H |
set (µ) is then an enumerative invariant of the complex manifold (V,J). Such numbers for
Σ,λ
H
algebraic manifolds (V,J), e.g. the complex projective spaces Pn, have been of great interest in
algebraic geometry for a long time.
If (V,ω,J) is a semipositive symplectic manifold, the symplectic invariant of (V,ω),
RT (;µ) RT (;µ ,... ,µ )
g,λ g,λ 1 N
≡
of [RT], is a well-defined integer. Due to the two composition laws of [RT], this symplectic invari-
ant is often more readily computable than the enumerative invariant (µ). In fact, all such
Σ,λ
|H |
symplectic invariants of Pn are easily computable. It is also shown in [RT] that the appropriately
defined genus-zero invariants of Pn agree with the corresponding symplectic invariants. On the
other hand, even for P2 and for genus one, the two invariants are no longer equal. In [I], the
difference
RT (µ ;µ ,... ,µ ) (µ)
1,λ 1 2 N Σ,λ
−|H |
is commuted for genus-one surfaces Σ and all projective spaces using an obstruction-bundle ap-
proach, first introduced by [T] in a very different setting. In [Z2], the difference
RT (;µ) (µ)
2,λ Σ,λ
−|H |
is commuted for genus-two surfaces Σ for P2 and P3 using a similar approach.
The main results of this paper are detailed descriptions of gluing pseudoholomorphic maps in sym-
plectic geometry. Theorem 3.29 concerns the adoption of the obstruction-bundle idea of [T] into
2
the symplectic setting. Theorem 3.33 describes the local structure of modulispaces of holomorphic
rational maps under certain regularity conditions. In addition to providing the necessary analytic
justification for the computations in [I] and [Z2], the two theorems lay foundations for compar-
ing enumerative and symplectic invariants in many Kahler manifolds, in particular, the complex
homogeneous manifolds. Furthermore, in [Z2], the second theorem is used to enumerate cuspidal
rational curves in P2 and two-component tacnodal rational curves in P3. The same arguments can
be used to count curves with higher-order degeneracies.
The author is grateful to T. Mrowka for pointing out the paper [I], encouraging the author to
work out all of the analytic issues arising in [I], and sharing some of his expertise in applications
of global analysis over countless hours of conversations. The author also thanks G. Tian for first
introducing him to Gromov’s symplectic invariants and helping him understand [LT].
1.2 Summary
Let Λ0,1π∗T∗Σ π∗TV Σ V denote the bundle of (J,j)-antilinear homomorphisms from
Σ ⊗ V −→ ×
π∗T∗Σ to π∗TV. If
Σ V
ν Γ Σ V;Λ0,1π∗T∗Σ π∗TV ,
∈ × Σ ⊗ V
denoteby Σ,ν,λ thesetofall smooth(cid:0) mapsufromΣtoPn sucht(cid:1)hatu∗[Σ]=λand∂¯uz=ν (z,u(z))
M | |
for all z Σ. If µ is an N-tuple of constraints as above, put
∈
(µ) = (Σ;y ,... ,y ;u): u ; y Σ, u(y ) µ i= 1,... ,N .
Σ,ν,λ 1 N Σ,ν,λ i i i
M ∈M ∈ ∈ ∀
If (V,ω,J) is semiposit(cid:8)ive, for generic choices of ν and µ, is a smooth finite-d(cid:9)imensional
Σ,ν,λ
M
oriented manifold, and (µ) is a zero-dimensional finite submanifold of ΣN, whose
Σ,ν,λ Σ,ν,λ
M M ×
cardinality (with sign) depends only the homology classes of λ ,... ,λ ; see [RT]. The symplectic
1 N
invariant RT (;µ) is the signed cardinality of the set (µ).
g,λ Σ,ν,λ
M
If ν 0 and (Σ;y ;u ) (µ), then a subsequence of (Σ;y ,u ) ∞ must converge
k ikC0 −→ i i ∈MΣ,νi,λ { i i }i=1
in the Gromov topology to one of the following:
(1) an element of (µ);
Σ,λ
H
(2) (Σ ;y;u), where Σ is a bubble tree of S2’s attached to Σ with marked points y ,... ,y , and
T T 1 N
u: Σ V is a holomorphic map such that u(y ) µ for i= 1,... ,N, and
T i i
−→ ∈
(2a) uΣ is simple and the tree contains at least one S2;
T
|
(2b) uΣ is multi-covered;
T
|
(2c) uΣ is constant and the tree contains at least one S2.
T
|
This convergence statement says that for all t sufficiently small all the elements of (µ) lie
Σ,tν,λ
M
near one of the spaces described by (1)-(2c). In many practical applications, it is easy to show that
there is a bijection between the elements of (µ) and the nearby elements of (µ); see
Σ,λ Σ,tν,λ
H M
Proposition 3.30. This is thecase for all projective spaces, provided λ is a sufficiently high multiple
of the line (depending on the genus g). In [I] and [Z2], Cases (2a) and (2b) do not occur, but
they may have to be considered when dealing with higher-dimensional projective spaces or higher
genera. Thus, the convergence statement implies that if the signed cardinality of (µ) is
Σ,tν,λ
M
RT (;µ) for all t>0 sufficiently small, the number of elements of (µ) that lie near the
g,λ Σ,tν,λ
M
spaces described by (2) is exactly RT (;µ) (µ) . The goal of this paper is to describe
2,λ Σ,λ
−|H |
this difference in terms of the spaces of holomorphic maps themselves, which can be viewed as an
enumerative object, rather than a symplectic one. We do need to assume that certain spaces of
3
holomorphic maps are smooth, but they do not need to have the expected dimension.
While there is a very good understanding of what constitutes a stable map, there is little in a way
of commonly accepted notation for stable maps and various spaces of stable maps. In Section 2,
we recall the definition of bubble or stable maps as well as set up analytically convenient notation.
Our notation for bubble maps evolved from that of D. McDuff’s lectures at Harvard. We also
define various spaces of bubble maps and restate the definition of the Gromov topology on the set
of all bubble maps in our notation.
In Section 3, we describe a gluing construction and an obstruction bundle setup. One difficulty
in the gluing procedure is that a certain operator has eigenvalues that tend to zero as the gluing
parameter tends to zero, but then disappear as the gluing parameter hits zero. This is not really
dealt with in [I], but there are now several standard approaches to this problem. We use the modi-
fied Sobolev norms of [LT], redefined in the notation of Section 2. The main goal of Section 3 is to
describe a correspondence between the elements of (µ) lying near the maps of type (2) and
Σ,tν,λ
M
the zero set of a map between two bundles; see Theorem 3.29. The domain and target bundles will
be the bundle of gluing parameters and the obstruction bundle, respectively. We also give a local
analytic description of spaces of stable rational maps into V under certain assumptions. If V =Pn,
these spaces are very familiar in algebraic geometry, but the description of Theorem 3.33 is used
for a transition from the analysis of the gluing problem to the topology of moduli spaces of stable
rational maps in [Z2].
Section 4 contains proofs of continuity, injectivity, and surjectivity of the gluing maps. These are
usually omitted in the literature, but in the given case one has to choose the obstruction bundle
setup carefully to ensure that these properties of the gluing map actually hold. In particular,
Section 4 contains what [LT] may mean by “asymptotic analysis near the nodes,” which they omit.
The Appendix deals with even more technical details of the analysis.
1.3 Fundamental Notation
In this subsection, we collect the most frequently used combinatorial and analytic notation.
Definition 1.1 A finite partially ordered set I is a linearly ordered set if for all i ,i ,h I such
1 2
∈
that i ,i <h, either i i or i i .
1 2 1 2 2 1
≤ ≤
A linearly ordered set I is a rooted tree if I has a unique minimal element, i.e. there exists ˆ0 I
∈
such that ˆ0 h for all h I.
≤ ∈
If I and I′ are linearly ordered sets, abijectionφ: I I′ isanisomorphism of linearly ordered sets
−→
if for all h,i I, i<h if and only if φ(i)<φ(h).
∈
Let I be a linearly ordered set. We denote the subset of the non-minimal elements of I by Iˆ, i.e.
Iˆ= h I:i < h for some i I .
{ ∈ ∈ }
For every h Iˆ, the set i I:i < h has a unique maximal element ι , i.e.
h
∈ { ∈ }
ι < h and i ι for all i I s.t. i < h.
h h
≤ ∈
For reasons made clear in Subsection 2.1, ι: Iˆ I will be called the attaching map of I. It is
−→
clear from Definition 1.1 that I has a unique splitting I = I such that I I is a rooted tree.
k k
⊂
k∈K
F
4
The attaching map of I restricts to the attaching map of each I , which will still be denoted by ι.
k
Let I be a rooted tree. We denote the unique minimal element of I by ˆ0 , or simply by ˆ0 if there
I
is no ambiguity. If I∗, I , and I are rooted trees, we will write Iˆ∗, Iˆ, and Iˆ for I∗, I , and I ,
∗ ∗ ∗
∗ ∗ ∗
respectively; here denotes any string of symbols. If i I, let
∗ ∈
b b c
D I = h I:h > i , D¯ I = D I i .
i i i
{ ∈ } ∪{ }
Every rooted treeI has anumberofsubsetsthatarerooted trees; thesubsetsD¯ I areoneexample.
i
If H is a subset of I, the set
I(H) i I: i> h h H
≡{ ∈ 6 ∀ ∈ }
is also a rooted tree. If i I, denote I({i}) by I(i). If H is a subset of Iˆ, let
∈
IH = i I:i h h H , I(H) = H ˆ0 .
I
{ ∈ 6≥ ∀ ∈ } ∪{ }
If h Iˆ, denote I{h} by Ih.
∈
If M and M are two sets, let M +M be the disjoint union of M and M . Finally, if N is a
1 2 1 2 1 2
nonnegative integer, let [N]= 1,... ,N .
{ }
We now introduce some analytic notation. Let β: R [0,1] be a smooth function such that
−→
0, if t 1;
β(t) = ≤ and β′(t)> 0 if t (1,2). (1.3)
(1, if t 2, ∈
≥
If r > 0, let βr C∞(R;R) be given by βr(t) = β(r−12t). Note that
∈
supp(βr)= [r12,2r12], kβr′kC0 ≤ Cβr−21, and kβr′′kC0 ≤ Cβr−1. (1.4)
Throughout the paper, β and β will refer to these smooth bump functions.
r
Let q ,q : C S2 R3 be the stereographic projections mapping the origin in C to the north
N S
−→ ⊂
and south poles, respectively. Explicitly,
2z 1 z 2 2z 1+ z 2
q (z) = , −| | C R, q (z) = ,− | | . (1.5)
N 1+ z 2 1+ z 2 ∈ × S 1+ z 2 1+ z 2
(cid:18) | | | | (cid:19) (cid:18) | | | | (cid:19)
We denote the south pole of S2, i.e. the point (0,0, 1) R3, by . Let
− ∈ ∞
∂
e = (0,0,1) = dq T S2, (1.6)
∞ S ∞
0 ∂s ∈
(cid:12) (cid:18) (cid:19)
(cid:12)
where we write z = s+it C. We identify C wit(cid:12)h S2 via the map q . If x,z S2 ,
N
∈ −{∞} ∈ −{∞}
define φ z C by
x
∈
φ z = z x q−1(z) q−1(x). (1.7)
x − ≡ N − N
Notethatthemapφ : S2 Cisbiholomorphic. Ifg isaRiemannianmetriconaRiemann
x
−{∞} −→
surface (Σ,j) of positive genus, x Σ and v T Σ, we write exp v Σ for the exponential
∈ ∈ x g,x ∈
5
of v defined with respect to the Levi-Civita connection of g. Let inj x denote the corresponding
g
injectivity radiusat x andd the distance function. If x,z Σ are such that d (x,z) < inj x, define
g g g
∈
φ z T Σ by
g,x x
∈
exp φ z = z, φ z < inj x. (1.8)
g,x g,x | g,x |g,x g
Note that if g is flat on a neighborhood U of x in Σ, then φ U is holomorphic.
g,x
|
Let g be the Kahler metric of (V,J,ω). Denote the corresponding Levi-Civita connection, expo-
V
nential map, and distance function by V, exp and d , respectively. For every λ H (V;Z), let
∇ V V ∈ 2
λ = ω,λ . The number λ is the g -energy of any element of ; see [MS]. By rescaling ω, it
V Σ,λ
| | h i | | H
can be assumed that λ 1, whenever λ = 0 and = . If g is any Kahler metric on (V,J),
S2,λ
| | ≥ 6 H 6 ∅
denote the corresponding Levi-Civita connection, exponential map, distance function, injectivity
radius, and the parallel transport along the geodesic for X TV by g, exp , d , inj , and Π ,
g g g g,X
∈ ∇
respectively. If q V and δ R, let B (q,δ) = q′ V : d (q,q′) δ . In our construction, we allow
g g
∈ ∈ ∈ ≤
g vary in a smooth family. Without causing any additional difficulty in the gluing construction,
(cid:8) (cid:9)
consideration of such families simplifies computations in specific cases such as in [Z2]. If (S,j) is a
smooth Riemann surface and u C∞(S;V), put
∈
Γ(u)= Γ(S;u∗TV), Γ1(u) = Γ(S;T∗S u∗TV);
⊗
1
Γ0,1(u) = Γ(S;Λ0,1T∗S u∗TV), ∂¯u = du+J du j Γ0,1(u).
⊗ 2 ◦ ◦ ∈
(cid:0) (cid:1)
WedenotebyD andD thelinearization of∂¯-operatorwithrespecttothemetricsg andg onV,
V g V
respectively. Since both metrics are Kahler, D and D commute with J and have no zeroth-order
V g
term; see [Z1].
It should be mentioned that it is not essential for the main gluing construction described in this
paper that (V,J,g) is Kahler or even symplectic. If (V,J,g) is not Kahler, we would need to choose
an orientation on certain spaces of holomorphic maps and take the induced orientation on the
cokernel bundle; see Subsection 3.2. Dropping the Kahler assumption would have almost no effect
on the analysis, but would slightly complicate the notation.
2 Spaces of Bubble Maps
2.1 Bubble Trees
LetS beeithertheRiemannsphereS2 orasmoothRiemannsurfaceΣofgenusatleast2. Allowing
the genus-one case would lead to somewhat more complicated notation, but would have no effect
on the analysis done in Section 3. Denote by S∗ the open subset S if S = S2 and S itself
−{∞}
if S = Σ.
Definition 2.1 A bubble tree based on S is a tuple = S,I;x , where
⊤
(1) I is a rooted tree and x: Iˆ S S2 is a map;
−→ ∪ (cid:0) (cid:1)
(2) x S∗ if ι =ˆ0 and x S2 otherwise;
h h h
∈ ∈ −{∞}
(3) if h = h and ι = ι , x = x .
1 6 2 h1 h2 h1 6 h2
6
Given a bubble tree as above, let Σ be the complex curve
⊤
⊤
Σ = ( ˆ0 S) ( h S2) ,
⊤
{ }× ⊔ { }× ∼
(cid:16) hG∈Iˆ (cid:17).
where (h, ) (ι ,x ) for h Iˆ. The subset Σ ˆ0 S of Σ will be called the principal
∞ ∼ h h ∈ ⊤,ˆ0 ≡ { }× ⊤
component of or Σ . If h Iˆ, let Σ = h S2. For every i I, Σ will be called the ith
⊤ ⊤,h ⊤,i
⊤ ∈ { }× ∈
bubble component of or Σ or simply a bubble component. Let Σ∗ and Σ∗ denote the open
⊤ ⊤ ⊤,i ⊤
subsets of smooth points of Σ and Σ , respectively, i.e.
⊤,i ⊤
Σ (i, ) (i,x ) :ι = i , if i Iˆ;
Σ∗ = ⊤,i−{ ∞ }−{ h h } ∈ Σ∗ = Σ∗ .
⊤,i (S −{(ˆ0,xh): ιh = ˆ0}, if i= ˆ0; ⊤ i[∈I ⊤,i
The complement of Σ∗ in Σ are the singular points of Σ .
⊤,i ⊤ ⊤
For every i I, let (i)= S,I(i);x Iˆ(i) . Similarly, for each h Iˆ, let h= S,Ih;xIˆh . These tu-
∈ ⊤ | ∈ ⊤ |
ples are again bubble trees based on S. The complex curve Σ is obtained from Σ by dropping
(cid:0) (cid:1) ⊤(i) (cid:0) ⊤ (cid:1)
all bubble components descendent from the ith bubble component. The curve Σ is obtained by
⊤h
dropping the hth bubble component along with all bubble components descendent from it.
If S=S2 and h Iˆ, we denote the map φ defined in (1.7) by φ . If z Σ , put
∈ xh ⊤,h ∈ ⊤,i
φ z , if i = ι and z = ;
⊤,h h
r (z) = | | 6 ∞ (2.1)
⊤,h
(100, otherwise.
If δ > 0, let B (δ) = z Σ :r (z) < δ . Put
⊤,h ⊤ ⊤,h
{ ∈ }
r =min q−1(x ) ,min r (ι ,x ): l = h . (2.2)
⊤ h∈Iˆ S h { ⊤,h l l 6 }
(cid:16)(cid:12) (cid:12) (cid:17)
(cid:12) (cid:12)
If S=Σ and h Iˆis such that ι Iˆ, we again let φ denote the function φ of (1.7) and define
∈ h∈ ⊤,h xh
r and B (δ) as above. If g is a Riemannian metric on Σ, ι = ˆ0, and z Σ , put
⊤,h ⊤,h h ⊤,i
∈
d (x ,z), if i = ˆ0;
g h
r (z) = (2.3)
⊤,g,h
(100, otherwise.
We denote by φ the function φ of (1.8) and by B (δ) the ball B (x ,δ). Put
⊤,g,h g,xh ⊤,g,h g h
r g = min min r (ι ,x ): l = h ,min q−1(x ) ,min r (ι ,x ) :l = h . (2.4)
⊤ ιh=ˆ0{ ⊤,g,h l l 6 } ιh6=ˆ0 S h { ⊤,h l l 6 }
(cid:16) (cid:0)(cid:12) (cid:12) (cid:1)(cid:17)
(cid:12) (cid:12)
We say g is a -admissible Riemannian metric on Σ if there exists δ > 0 such that for all h Iˆ
⊤ ∈
with ι = ˆ0 the metric g is flat on B (δ).
h ⊤,g,h
2.2 The Basic Gluing Construction
In this subsection, we describe a gluing construction on which is the basis of all the other gluing
⊤
constructions in this paper. Lemma 2.2 plays a very important role in the next section and in the
7
explicit computations of [Z2].
Let = S,I;x be a bubble tree. If h Iˆ, put
⊤ ∈
(cid:0) (cid:1) C, if x S2;
(0) h (0) (0)
F = ∈ F = F . (2.5)
h,⊤ (TxhΣ, if xh ∈ Σ, ⊤ Mh∈Iˆ h,⊤
If S=S2, for any δ > 0, put
(0) (0)
F = υ = ( ,v ): v F , υ v < δ .
⊤,δ ⊤ Iˆ Iˆ∈ ⊤ | | ≡ | h|
n Xh∈Iˆ o
1
Let δ (0,1) be such that 8δ2 < r . If S = Σ and g is an admissible metric on Σ, put
⊤∈ ⊤ ⊤
(0) (0)
F = υ = ( ,v ): v F , υ v + v < δ ,
⊤,g,δ ⊤ Iˆ Iˆ∈ ⊤ | |g ≡ | h|g | h|
n iXh=ˆ0 iXh6=ˆ0 o
1
where |vh|g = |vh|g,xh. Let δ⊤g ∈(0,1) be such that 8(δ⊤g)2 <r⊤g and the metric g is flat on
Bg xh,4(δ⊤g)12 for all h Iˆwith ιh=ˆ0. We now construct a family of bubble trees parameterized
∈
by F(0) with δ=δ if S=S2 and by F(0) with δ=δ g if S=Σ.
(cid:0) ⊤,δ (cid:1) ⊤ ⊤,g,δ ⊤
First, for every h Iˆand v F(0) with
∈ h∈ ⊤,h
v (0,δ) if x S2 and v (0,δ) if x Σ,
h h h g h
| |∈ ∈ | | ∈ ∈
we define local stretching maps
q :Σ Σ if x S2 and q :Σ Σ if x Σ
h,(xh,vh) ⊤(ιh) −→ ⊤(h) h∈ g,h,(xh,vh) ⊤(ιh) −→ ⊤(h) h∈
as follows. If xh∈S2, let ph,(xh,vh):B⊤,h(δ12) −→ C∪∞ be given by
v
h
p (z) = 1 β (2φ z ) . (2.6)
h,(xh,vh) − |vh| | ⊤,h | φ z
(cid:18) ⊤,h (cid:19)
(cid:0) (cid:1)
Define q : Σ Σ by
h,(xh,vh) ⊤h −→ ⊤(h)
h,qS(ph,(xh,vh)(z)) , if 12 ≤ |vh|−12r⊤,h(z) ≤ 1;
qh,(xh,vh)(z) = (cid:0)ιh,φ−⊤1,h β|vh|(|φ⊤,(cid:1)hz|)(φ⊤,hz) , if 1 ≤ |vh|−21r⊤,h(z) ≤ 2; (2.7)

(cid:16)z, (cid:0) (cid:1)(cid:17) otherwise.

Note that qh,(xh,vh) issmooth andis adiffeomorphism,except on thecircle r⊤,h(z) = |vh|12 inΣ⊤,ιh.
1
The map qh,(xh,vh) stretches B⊤,h(|vh|2) around the sphere Σ⊤,h. If xh∈Σ, similarly to the above,
let pg,h,(xh,vh):B⊤,g,h(δ12)−→ C∪∞ be given by
v
h
p (z) = 1 β (2φ z ) . (2.8)
g,h,(xh,vh) − |vh|g | ⊤,g,h |g φ z
(cid:18) ⊤,g,h (cid:19)
(cid:0) (cid:1)
8
Note that the ratio v /φ z is well-defined as an extended complex number, since T Σ is one-
h ⊤,g,h xh
dimensional and v = 0. Define q : Σ Σ by
h 6 g,h,(xh,vh) ⊤h −→ ⊤(h)
h,q (p (z)) , if 1 v −12r (z) 1;
S g,h,(xh,vh) 2 ≤ | h|g ⊤,g,h ≤
qg,h,(xh,vh)(z) = (cid:0)ιh,φ−g,1⊤,h β|vh|g(|φ⊤(cid:1),g,hz|g)(φ⊤,g,hz) , if 1 ≤ |vh|−g 12r⊤,g,h(z) ≤ 2; (2.9)

z(cid:16), (cid:0) (cid:1)(cid:17) otherwise.


Similarly to the case x S2, q is smooth and is a diffeomorphism, except on the circle
h∈ g,h,(xh,vh)
1
r (z) = v 2 in Σ .
⊤,g,h | h|g ⊤,ih
If S =S2, for every h I and υ F(0), we now define a bubble tree (υ) and a smooth map
∈ ∈ ⊤,δ ⊤h
q : Σ Σ . Choose an ordering of I consistent with its partial ordering. If h=ˆ0, we
υ,h ⊤h(υ) −→ ⊤(h)
take I (υ) = ˆ0 , (υ)= S,I (υ); , and q = Id . Suppose h=ˆ0 and
h h h υ,h S
{ } ⊤ 6
(cid:0) (cid:1) (υ) = S,Iˆ (υ);x (υ)
h−1 h−1 h
⊤
(cid:0) (cid:1)
with I (υ) I. If v = 0, put
h−1 h
⊂
ι (υ),x (υ) , if l I (υ);
h−1,l h−1,l h−1
I (υ) = I (υ) h , ι (υ),x (υ) = ∈
h h−1 ∪{ } h,l h,l (q(cid:0)υ−,1ιh(ιh,xh), (cid:1) otherwise.
(cid:0) (cid:1)
Let q Σ = q and q (h,z) = (h,z). If v = 0, let
υ,h| ⊤h−1(υ) υ,h−1 υ,h h 6
I (υ) = I (υ), ι (υ),x (υ) = ι (υ),x (υ) .
h h−1 h,l h,l h−1,l h−1,l
(cid:0) (cid:1) (cid:0) (cid:1)
Take q = q q . Inductively this procedure defines a bubble tree (υ) = (υ)
υ,h h,(xh,vh) ◦ υ,h−1 ⊤ ⊤h∗
based on S and a smooth map q = q : Σ Σ , which is a diffeomorphism outside of
υ υ,h∗ ⊤(υ) ⊤
−→
I I(υ) disjoint circles, where h∗ is the largest element of I. The resulting bubble tree and map
| − |
are independent of the choice of the extension of the partial ordering. While the domains of the
maps q do depend on such a choice, whenever we make use of the maps q below, the result
υ,h υ,h
(0)
will also be independent of the choice. If S=Σ, for every h I and υ F , we define bubble
∈ ∈ ⊤,g,δ
tree (υ) and maps q : Σ Σ similarly to the above, but replacing q by
⊤g,h g,υ,h ⊤g,h(υ) −→ ⊤(h) h,(xh,vh)
q whenever ι =ˆ0. We let (υ)= (υ) and q =q . As before, q is smooth
g,h,(xh,vh) h ⊤g ⊤g,h∗ g,υ g,υ,h∗ g,υ
and a diffeomorphism outside of I I(υ) disjoint circles.
| − |
If S=S2 and v =0, put
h
6
A+υ,h = qυ−,1ιh z∈Σ⊤,ιh: |vh|12 ≤ r⊤,h(z) ≤ 2|vh|21 ;
(2.10)
A−υ,h = qυ−,1ιh(cid:16)(cid:8)z∈Σ⊤,ιh: 12|vh|21 ≤ r⊤,h(z) ≤ |vh|21(cid:9)(cid:17).
(cid:16) (cid:17)
(cid:8) (cid:9)
Note that A± Σ , where
υ,h ⊂ ⊤(υ),i∗h(υ)
i∗(υ) = min i I: i< h and v = 0 if i < h′ < h = max i I(υ): i < h .
h { ∈ h′ 6 } { ∈ }
9
If S=Σ and v =0, we similarly define
h
6
1 1
A+ = q−1 z Σ : v 2 r (z) 2v 2 ;
g,υ,h g,υ,ιh ∈ ⊤,ιh | h|g ≤ ⊤,g,h ≤ | h|g
(2.11)
A− = q−1 (cid:16)(cid:8)z Σ : 1 v 12 r (z) v 21(cid:9)(cid:17),
g,υ,h g,υ,ιh ∈ ⊤,ιh 2| h|g ≤ ⊤,g,h ≤ | h|g
(cid:16) (cid:17)
(cid:8) (cid:9)
where v and r denote v and r if ι Iˆ.
h g ⊤,g,h h ⊤,h h
| | | | ∈
Lemma 2.2 If S = S2, the map q is holomorphic outside of the annuli A± with v = 0.
υ υ,h h 6
For such h,
dq C;
k h,(xh,vh)kC0(qυ,ιh(A±υ,h)) ≤
v
∂¯ qυ ◦qυ−,1ιh z = −2|vh|−12 (cid:18)φ⊤,hhz(cid:19)dqS ph,(xh,vh)z ◦∂β 2|vh|−12φ⊤,hz ◦dφ⊤,h z ∀z∈qυ,ιh(A−υ,h),
(cid:0) (cid:1)(cid:12) (cid:12) (cid:12) (cid:12)
where the norm(cid:12) is computed with respect(cid:12)to the standar(cid:12)d metric on S2, an(cid:12)d β is a viewed as a
function on C via the standard norm on C. If S=Σ, the map q is holomorphic outside of the
g,υ
annuli A± with v =0. For such h,
g,υ,h h6
dq C if ι = ˆ0; dq C if ι = ˆ0;
k g,h,(xh,vh)kC0(qg,υ,ιh(A±g,υ,h)) ≤ g h k h,(xh,vh)kC0(qg,υ,ιh(A±g,υ,h)) ≤ h 6
v
∂¯ qg,υ ◦qg−,υ1,ιh z = −2|vh|−12 (cid:18)φ⊤h,hz(cid:19)dqS ph,(xh,vh)z ◦∂β 2|vh|−12φ⊤,hz ◦dφ⊤,h z ∀z∈qg,υ,ιh(A−υ,h),
(cid:0) (cid:1)(cid:12) (cid:12) (cid:12) (cid:12)
where we regard(cid:12) β as a function on T Σ v(cid:12)ia the metric g(cid:12) and denote φ (cid:12)by φ if ι =ˆ0.
xh g,⊤,h ⊤,h h
Proof: The first statement in each of the two cases is immediate from the construction. The
estimates on the differential of q and q follow from (1.4). Suppose S=Σ, ι =ˆ0,
h,(xh,vh) g,h,(xh,vh) h
v = 0 and z A− . Since q = q on A− and q is anti-holomorphic, from (2.8)
h 6 ∈ g,υ,h g,υ g,h,(xh,vh) g,υ,h S
and (2.9) we obtain
∂¯q = dq ∂p
g,υ z S pg,h,(xh,vh)z ◦ g,h,(xh,vh) z
(cid:12)(cid:12) = 2(cid:12)(cid:12)v −12 vh dq (cid:12)(cid:12) ∂β dφ .
− | h|g (cid:18)φ⊤,g,hz(cid:19) S pg,h,(xh,vh)z ◦ 2|vh|−g 12φ⊤,g,hz ◦ ⊤,g,h z
(cid:12) (cid:12) (cid:12)
The other cases are proved similarly, since q (cid:12) q−1 = q (cid:12) on q (A− )(cid:12)and a similar
g,υ ◦ g,υ,ιh h,(xh,vh) g,υ,ιh g,υ,h
statement holds in the case S = S2.
2.3 Curves with Marked Points
Definition 2.3 If M is a finite set, a curve with M-marked points based on S is a tuple
= S,M,I;x,(j,y) , where
C
(1) S,I;x is a bubble tree bas(cid:0)ed on S, and j:(cid:1)M I and y:M S S2 are maps;
C
⊤ ≡ −→ −→ ∪
(2) j I, (j ,y ) Σ∗ , and y = for all l M;
l∈ (cid:0) l l (cid:1)∈ ⊤C,jl l 6 ∞ ∈
(3) for any l ,l M with l = l and j = j , y = y .
1 2∈ 1 6 2 l1 l2 l1 6 l2
The curve is stable if h: ι = i + l: j = i 2 for all i Iˆif S=Σ and all i I if S=S2.
h l
C { } { } ≥ ∈ ∈
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) 10(cid:12)