Chiu-Chu Melissa Liu- Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open...

8/3/2019 Chiu-Chu Melissa Liu- Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witt

1/85

arXiv:math/0210257v2

[math.S

G]4Dec2004

MODULI OF J-HOLOMORPHIC CURVES WITH LAGRANGIANBOUNDARY CONDITIONS AND OPEN GROMOV-WITTEN

INVARIANTS FOR AN S1-EQUIVARIANT PAIR

CHIU-CHU MELISSA LIU

Abstract. Let (X, ) be a symplectic manifold, J be an -tame almost com-plex structure, and L be a Lagrangian submanifold. The stable compactifi-

cation of the moduli space of parametrized J-holomorphic curves in X withboundary in L (with prescribed topological data) is compact and Hausdorff inGromovs C-topology. We construct a Kuranishi structure with corners in

the sense of Fukaya and Ono. This Kuranishi structure is orientable if L isspin. In the special case where the expected dimension of the moduli space is

zero, and there is an S1 action on the pair (X,L) which preserves J and actsfreely on L, we define the Euler number for this S1 equivariant pair and theprescribed topological data. We conjecture that this rational number is the

one computed by localization techniques using the given S1 action.

Contents

1. Introduction 21.1. Background 21.2. Main results 32. Surfaces with Analytic or Dianalytic Structures 5

2.1. Analyticity and dianalyticity 52.2. Various categories of surfaces 62.3. Topological types of compact symmetric Riemann surfaces 83. Deformation theory of Bordered Riemann Surfaces 83.1. Deformation theory of smooth bordered Riemann surfaces 83.2. Nodal bordered Riemann surfaces 123.3. Deformation theory for prestable bordered Riemann surfaces 174. Moduli of Bordered Riemann Surfaces 234.1. Various moduli spaces and their relationships 234.2. Decomposition into pairs of pants 254.3. Fenchel-Nielsen coordinates 264.4. Compactness and Hausdorffness 294.5. Orientation 31

5. Moduli Space of Stable Maps 325.1. Prestable and stable maps 325.2. C topology 335.3. Compactness and Hausdorffness 356. Construction of Kuranishi Structure 366.1. Kuranishi structure with corners 36

Date: February 1, 2008.

1
http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2http://arxiv.org/abs/math/0210257v2


2/85

2

6.2. Stable Wk,p maps 386.3. Deformation of the domain 456.4. Local Charts 516.5. Transition functions 706.6. Orientation 717. Virtual fundamental chain 737.1. Construction of virtual fundamental chain 737.2. S1 action 787.3. Invariants for an S1-equivariant pair 807.4. Multiple covers of the disc 82References 83

1. Introduction

1.1. Background. String theorists have been making predictions on enumerativeinvariants using dualities. One of the most famous examples is the astonishingpredictions of the number of rational curves in a quintic threefold in [CdGP]. Tounderstand these predictions, mathematicians first developed Gromov-Witten the-ory to give the numerical invariants a rigorous mathematical definition so thatthese predictions could be formulated as mathematical statements, and then triedto prove these statements. The predictions in [CdGP] are proven in [Gi, LLY].

Recently, string theorists have produced enumerative predictions about holomor-phic curves with Lagrangian boundary conditions by studying dualities involvingopen strings [OV, LMV, AV, AKV, MV]. Moreover, assuming the existence of avirtual fundamental cycle and the validity of localization formulas, mathematicianshave carried out computations which coincide with these predictions [KL, GZ]. In

certain cases, these numbers can be reproduced by considering relative morphisms[LS]. It is desirable to give a rigorous mathematical definition of these enumera-tive invariants, so that we may formulate physicists predictions as mathematicaltheorems, and then try to prove these theorems.

Gromov-Witten invariants count J-holomorphic maps from a Riemann surfaceto a fixed symplectic manifold (X, ) together with an -tame almost complexstructure J. These numbers can be viewed as intersection numbers on the modulispace of such maps. We want the moduli space to be compact without boundaryand oriented so that there exists a fundamental cycle which allows us to do inter-section theory. The moduli space of J-holomorphic maps can be compactified byadding stable maps, whose domain is a Riemann surface which might have nodalsingularities. The stable compactification is compact and Hausdorff in the C

topology defined by Gromov [Gr]. The moduli space is essentially almost complex,

so it has a natural orientation. In general, the moduli space is not of the expecteddimension and has bad singularities, but there exists a virtual fundamental cyclewhich plays the role of fundamental cycles [LT1, BF, FO, LT2, Sie1]. These arenow well-established in Gromov-Witten theory.

The open Gromov-Witten invariants that I want to establish shall count J-holomorphic maps from a bordered Riemann surface to a symplectic manifold Xas above such that the image of the boundary lies in a Lagrangian submanifoldL of X. To compactify the moduli space of such maps, Sheldon Katz and I [KL]


3/85

3

introduced stable maps in this context. The stable compactification is compact andHausdorff in the C topology, as in the ordinary Gromov-Witten theory. However,orientation is a nontrivial issue in the open Gromov-Witten theory. Moreover, theboundary is of real codimension one, so the compactified moduli space does notclose up as in the ordinary Gromov-Witten theory, and the best we can expectis a fundamental chain.

A fundamental chain is not satisfactory for intersection theory. For example,the Euler characteristic of a compact oriented manifold without boundary can bedefined as the number of zeros of a generic vector field, counted with signs deter-mined by the orientation. This number is independent of the choice of the vectorfield, and thus well-defined. For a compact oriented manifold with boundary, onecan still count the number of zeros of a generic vector field with signs determinedby the orientation, but the number will depend on the choice of the vector field.Therefore, we need to specify extra boundary conditions to get a well-defined num-ber.

1.2. Main results. Let (X, ) be a symplectic manifold of dimension 2N, and Lbe a Lagrangian submanifold. To compactify the moduli space of parametrizedJ-holomorphic curves in X with boundary in L, Gromov introduced cusp curveswith boundary [Gr], which I will call prestable maps. A prestable map to (X, L)

is a continuous map f : (, ) (X, L) such that f : (, ) (X, L) isJ-holomorphic, where is a prestable (i.e., smooth or nodal) bordered Riemann

surface, and : is the normalization map [KL, Definition 3.6.2].A smooth bordered Riemann surface is of type (g, h) if it is topologically a

sphere with g handles and h holes. The boundary of consists of h disjoint cir-cles B1, . . . , Bh. We say has (n, m) marked points if there are n distinct markedpoints in its interior and mi distinct marked points on Bi, where m = (m1, . . . , mh),mi 0. By allowing nodal singularities, we have the notion of a prestable Riemannsurface of type (g, h) with (n, m) marked points and an ordering B

1

, . . . , Bh

of theboundary components. An isomorphism between two such prestable bordered Rie-mann surfaces is an isomorphism of prestable bordered Riemann surfaces whichpreserves the marked points and ordering of boundary components. An isomor-phism between two prestable maps f : X and f : X is an isomorphism : in the above sense such that f = f . A prestable map is stable ifits automorphism group is finite. This is the analogue of Kontsevichs stable maps[Ko] in the ordinary Gromov-Witten theory.

For H2(X, L;Z), = (1, . . . , h) H1(L;Z)h, and Z, defineM(g,h),(n,m)(X, L | , , )

to be the moduli space of isomorphism classes of stable maps f : (, ) (X, L), where is a prestable bordered Riemann surface of type (g, h) with (n, m)

marked points and an ordering B1

, . . . , Bh

of the boundary components, f[] = ,f[Bi] = i, i = 1, . . . , h, and (fTX , (f|)TL) = . Here (fTX , ((f|)TL)is the Maslov index defined in [KL, Definition 3.3.7].

Theorem 1.1. M(g,h),(n,m)(X, L | , , ) is compact and Hausdorff in the Ctopology.

Here the C topology is the one defined by Gromovs weak convergence [Gr].The stability condition is necessary for Hausdorffness. The compactness follows


4/85

4

from [Gr, Ye], which will be explained in Section 5.3. I do not claim any originalityfor Theorem 1.1.

The boundary of the moduli space corresponds to degeneration of the domain orblowup of the map. An interior node corresponds to a (real) codimension 2 stratum,while a boundary node corresponds to a codimension 1 stratum. Blowup of the mapat an interior point leads to the well-known phenomenon of bubbling off of sphereswhich is codimension 2, while blowup at a boundary point leads to bubbling off ofdiscs which is codimension 1. The intersection of two or more codimension 1 strataforms a corner. It is shown in Section 6 that

Theorem 1.2. M(g,h),(n,m)(X, L | , , ) has a Kuranishi structure with cornersof (real) virtual dimension + (N 3)(2 2g h) + 2n + m1 + + mh, where 2Nis the (real) dimension of X. The Kuranishi structure is orientable ifL is spin orif h = 1 and L is relatively spin (i.e., L is orientable and w2(TL) = |L for some H2(X,Z2)).

The case for the disc with only boundary marked points (g = n = 0, h = 1) isproven in [FO3]. I will describe briefly what a Kuranishi structure with corners isand refer to Section 6.1 for the complete definition. A chart of a Kuranishi structurewith corners is a 5-uple (V,E, , , s), where V is a smooth manifold (possibly withcorners), is a finite group acting on V, E is a -equivariant vector bundle over V,s : V E is a -equivariant section, and maps s1(0)/ homeomorphically toan open set of the moduli. The dimension of V, rank of E, and the finite group might vary with charts, but d = dim VrankE is fixed and is the virtual dimensionof the Kuranishi structure with corners. det(T V) (det E)1 can be glued to anorbibundle, the orientation bundle, and the Kuranishi structure with corners isorientable if its orientation bundle is a trivial real line bundle.

If s intersects the zero section transversally, s1(0) is a manifold (possibly with

corners) of dimension d. In general, s1

(0) might have dimension larger than dand bad singularities due to the nontransversality of s. The virtual fundamentalchain can be constructed by perturbing s to a transversal section. Locally it is asingular chain with rational coefficients in V/ which is a rational combination ofthe images of d-dimensional submanifolds of V.

A virtual fundamental chain is not satisfactory for intersection theory. For ex-ample, when X is a Calabi-Yau threefold and L is a special Lagrangian submani-fold, M(g,h),(0,0)(X, L | , , ) is empty for = 0, and the expected dimension ofM(g,h),(0,0)(X, L | , , 0) is zero for any g, h, , . The virtual fundamental chainis a zero chain with rational coefficients, and we would like to define the invariant(g,h)(X, L | , , ) Q to be the degree of this zero chain. However, this numberdepends on the perturbation, so we need to impose extra boundary conditions toobtain a well-defined number. Now assume that

There is an S1 action : S1 X X which preserves J and L. The restriction of to L is free. The virtual dimension of M(g,h),(0,0)(X, L | , , ) is zero.

Under the above assumptions I can, using the S1 action , impose boundary con-ditions to get a well-defined rational number

(g,h)(X, L,

| , , )


5/85

5

which is an invariant of the equivariant pair (X, L,

), but not an invariant ofthe pair (X, L). I conjecture that these rational numbers are the ones computedby localization techniques using the S1 action

[KL, GZ]. The computations in

[KL, LS, GZ] coincide with physicists predictions.

Acknowledgments: First and foremost, I would like to thank my thesis advisorShing-Tung Yau for leading me to the field of mirror symmetry and providing thebest environment to learn its newest developments. Secondly, I thank CliffordTaubes and Gang Liu for answering my questions on symplectic geometry andcarefully reading the draft. I thank Sheldon Katz for being so generous to cooperatewith me. The cooperation [KL] is a very instructive experience for me and led meto this project. I thank Cumrun Vafa for suggesting to me this fruitful problem andpatiently explaining his works to me. I thank Jason Starr for being an incrediblygenerous and patient mentor of algebraic geometry. I thank Xiaowei Wang foralways being a source of mathematical knowledge and moral support. I thankArthur Greenspoon for his numerous valuable suggestions on the first draft, andChien-Hao Liu for his meticulous proofreading. I thank Spiro Karigiannis for hisgreat help on my English. In addition, it is a pleasure to thank William Abikoff,Selman Akbulut, Raoul Bott, Kevin Costello, Yong Fu, Kenji Fukaya, Tom Graber,Irwin Kra, Kefeng Liu, Curtis McMullen, Maryam Mirzakhani, Yong-Geun Oh,Kaoru Ono, Scott Wolpert, and Eric Zaslow for helpful conversations. Finally,special thanks go to Ezra Getzler for corrections and refinements of the part onmoduli spaces of bordered Riemann surfaces.

2. Surfaces with Analytic or Dianalytic Structures

In this section, we review some definitions and facts of surfaces with analytic ordianalytic structures, following [AG, Chapter 1] closely. This section is an expansionof Section 3.1 and 3.2 of [KL].

The marked bordered Riemann surfaces defined in Section 2.2.5 are directly re-lated to open Gromov-Witten theory.

2.1. Analyticity and dianalyticity.

Definition 2.1. Let A be a nonempty open subset ofC, f : A C a map. f isanalytic on A if fz = 0, and antianalytic onA if

fz = 0. f is said to be dianalytic

if its restriction to each component of A is either analytic of antianalytic.

Definition 2.2. LetA and B be nonempty subsets ofC+ = {z C | Imz 0}. Acontinuous function f : A B is analytic (resp. antianalytic on A if it extendsto an analytic (resp. antianalytic) function fC : U C, where U is an openneighborhood of A inC. f is said to be dianalytic on A if its restriction to eachcomponent of A is either analytic or antianalytic.

Theorem 2.3 (Schwarz reflection principle). Let A and B be nonempty subsetsof C+ = {z C | Imz 0}. A continuous function f : A B is dianalytic(resp. analytic) if it is dianalytic (resp. analytic) on the interior ofA and satisfiesf(A R) R.Definition 2.4. A surface is a Hausdorff, connected, topological space togetherwith a family A = {(Ui, i) | i I} such that {Ui | i I} is an open covering of and each map i : Ui Ai is a homeomorphism onto an open subset Ai ofC+.


6/85

6

A is called a topological atlas on , and each pair (Ui, i) is called a chart of A.The boundary of is the set

= {x | i I s.t. x Ui, i(x) R}ij i 1j : j(Ui Uj) i(Ui Uj) are surjective homeomorphisms, calledthe transition functions of A. A is called a dianalytic (resp. analytic) atlas if allits transition functions are dianalytic (resp. analytic).

2.2. Various categories of surfaces.

2.2.1. Riemann surfaces.

Definition 2.5. A Riemann surface is a surface equipped with the analytic structureinduced by an analytic atlas on it.

A Riemann surface is canonically oriented by its analytic structure.

2.2.2. Symmetric Riemann surfaces.

Definition 2.6. A symmetric Riemann surface is a Riemann surface togetherwith an antiholomorphic involution : . is called the symmetry of .Definition 2.7. A morphism between symmetric Riemann surfaces (, ) and(, ) is an analytic map f : such that f = f.Definition 2.8. A symmetric Riemann surface with (n, m) marked points is a sym-metric Riemann surface (, ) together with2n+m distinct points p1, . . . , p2n+m in such that (pi) = pn+i for i = 1, . . . , n and (pi) = pi fori = 2n + 1, . . . , 2n + m.

2.2.3. Klein surfaces.

Definition 2.9. A Klein surface is a surface equipped with the dianalytic structureinduced by a dianalytic atlas on it.

A Riemann surface can be viewed as a Klein surface. A Klein surface can beequipped with an analytic structure compatible with the dianalytic structure if andonly if it is orientable. In particular, an orientable Klein surface without boundaryadmits a compatible structure of a Riemann surface.

Definition 2.10. A morphism between Klein surfaces and is a continu-ous map f : (, ) (, ) such that for any x there exist analyticcharts (U, ) and (V, ) about x and f(x) respectively, and an analytic functionF : (U) C such that the following diagram commutes:

Uf V

(U) F C C+where (x + iy) = x + i|y| is the folding map.

Given a Klein surface , there are three ways to construct an unramified doublecover of . We refer to [AG, 1.6] for the precise definition of an unramified doublecover and detailed constructions. The complex double C is an orientable Kleinsurface without boundary. The orienting double O is an orientable Klein surface.It is disconnected if and only if is orientable, and it has nonempty boundary if


7/85

7

and only of does. The Schottkey double S is a Klein surface without boundary.It is disconnected if and only if the boundary of is empty, and it is nonorientableif and only if is.

If is orientable, then C = S, and O is disconnected (the trivial doublecover). If = , then C = O, and S is disconnected. In particular, if comes from a Riemann surface, then all three covers are the trivial disconnecteddouble cover.

Example 2.11. Let be a Mobius strip. ThenC is a torus, S is a Klein bottle,and O is an annulus.

2.2.4. Bordered Riemann surfaces.

Definition 2.12. A bordered Riemann surface is a compact surface with nonemptyboundary equipped with the analytic structure induced by an analytic atlas on it.

Remark 2.13. A bordered Riemann surface is canonically oriented by the analytic

(complex) structure. In the rest of this paper, the boundary circles Bi

of a borderedRiemann surface with boundary = B1 . . . Bh will always be given theorientation induced by the complex structure, which is a choice of tangent vector toBi such that the basis (the tangent vector ofBi, inner normal) for the real tangentspace is consistent with the orientation of induced by the complex structure.

Definition 2.14. A morphism between bordered Riemann surfaces and isa continuous map f : (, ) (, ) such that for any x there existanalytic charts (U, ) and (V, ) about x and f(x) respectively, and an analytic

function F : (U) C such that the following diagram commutes:U

f V

(U) F CA bordered Riemann surface is topologically a sphere with g 0 handles and

with h > 0 discs removed. Such a bordered Riemann surface is said to be of type(g, h).

A bordered Riemann surface can be viewed as a Klein surface. Its complexdouble and Schottkey double coincide since it is orientable.

2.2.5. Marked bordered Riemann surfaces. The following refinement of an earlierdefinition is suggested to the author by Ezra Getzler.

Definition 2.15. Leth be a positive integer, g, n be nonnegative integers, and m =(m1, . . . , mh) be an h-uple of nonnegative integers. A marked bordered Riemannsurface of type (g, h) with (n, m) marked points is an (h + 3)-uple

(, B; p; q1, . . . , qh)

whose components are described as follows.

is a bordered Riemann surface of type (g, h). B = (B1, . . . , Bh), where B1, . . . , Bh are connected components of, ori-

ented as in Remark 2.13. p = (p1, . . . , pn) is an n-uple of distinct points in . qi = (qi1, . . . , qimi) is an mi-uple of distinct points on the circle Bi.


8/85

8

Let 0 = (0, . . . , 0). Note that a marked bordered Riemann surface of type (g, h)

with (n,0) marked points is a bordered Riemann surface togther with an ordering

of the h boundary components.Definition 2.16. A morphism between marked bordered Riemann surfaces of type(g, h) with (n, m) marked points

(, B; p; q1, . . . , qh) (, B; p; (q)1, . . . , (q)h)is an isomorphism of bordered Riemann surface f : such thatf(Bi) = (B)i

for i = 1, . . . , h, f(pj) = pj for j = 1, . . . , n, and f(q

ik) = (q

)ik for k = 1, . . . , mi.

Remark 2.17. The category of marked bordered Riemann surfaces of type (g, h)with (n, m) marked points is a groupoid since every morphism in Definition 2.16 isan isomorphism.

2.3. Topological types of compact symmetric Riemann surfaces. A com-pact symmetric Riemann surface is topologically a compact orientable surface with-

out boundary together with an orientation reversing involution , which is clas-sified by the following three invariants:

(1) The genus g of .(2) The number h = h() of connected components of , the fixed locus of .(3) The index of orientability, k = k() 2the number of connected compo-

nents of \.These invariants satisfy:

(1) 0 h g + 1.(2) For k = 0, h > 0 and h g + 1 (mod 2).(3) For k = 1, 0 h g.

The above classification was realized already by Felix Klein (see e.g. [Kl], [We],[Se]). This classification is probably better understood in terms of the quotient

Q() = /, where = {id,} is the group generated by . The quotientQ() is orientable if k = 0 and nonorientable if k = 1, hence the name indexof orientability. Furthermore, h is the number of connected components of theboundary of Q(). If Q() is orientable, then it is topologically a sphere withg 0 handles and with h > 0 discs removed, and the invariants of (, ) are(g ,h,k) = (2g + h 1, h, 0). IfQ() is nonorientable, then it is topologically asphere with g > 0 crosscaps and with h 0 discs removed, and the invariants of are (g , h, k) = (g + h 1, h, 1).

From the above classification we see that symmetric Riemann surfaces of a givengenus g fall into [ 3g+42 ] topological types.

3. Deformation theory of Bordered Riemann Surfaces

In this section, we study deformation theory of bordered Riemann surfaces. Werefer to [KL, Section 3] for some preliminaries such as doubling constructions andthe Riemann-Roch theorem for bordered Riemann surfaces.

3.1. Deformation theory of smooth bordered Riemann surfaces. Let bea bordered Riemann surface, (C, ) be its complex double (see e.g. [KL, Section3.3.1] for the definition). Analytically, (C, ) is a compact symmetric Riemannsurface. Algebraically, it is a smooth complex algebraic curve X which is thecomplexification of some smooth real algebraic curve X0, i.e., X = X0 R C (see


9/85

9

[Ha, Chapter II, Exercise 4.7]). Alternatively, (X, S) is a complex algebraic curvewith a real structure (see [Sil, I.1]), where S is a semi-linear automorphism in thesense of[Ha, Chapter II, Exercise 4.7] which induces the antiholomorphic involution on C.

3.1.1. Algebraic approach. First order deformations of the complex algebraic curveX is canonically identified with the complex vector space Ext1OX (X , OX), where1X is the sheaf of Kahler differentials on X. The obstruction lies in Ext

2OX (X , OX) =

0. Similarly, the first order deformation of the real algebraic curve X0 is identifiedwith the real vector space Ext1OX0 (X0 , OX0), and the obstruction vanishes. Wehave

Ext1OX (X , OX) = Ext1OX0 (X0 , OX0) R Csince X = X0RC. The semi-linear automorphism Sinduces a complex conjugation

S : Ext1OX (X , OX) Ext1OX (X , OX).The fixed locus Ext1OX (X , OX)S gives the first order deformation of (X, S) asa complex algebraic curve with a real structure, and is naturally isomorphic toExt1OX0 (X0 , OX0).

More explicitly, X can be covered by complex affine curves which is a completeintersection of hypersurfaces defined by polynomials with real coefficients. Defor-mation of X are given by varying the coeffients (in C). Deformation of (X, S) isgiven by varying the coeffients in R. The above polynomials with real coefficientsalso define the real algebraic curve X0, and varying the coeffients in R gives the de-formation of X0. The complex conjugation of coefficients corresponds to the abovecomplex conjugation S on Ext1OX (X , OX).

X is a smooth algebraic variety, so Ext1OX (X , OX) is isomorphic to the sheafcohomology group H1(X, X), where X is the tangent sheaf of X, and

Ext2OX (X , OX) = H2(X, X) = 0.Similarly, we have

Ext1OX0 (X0 , OX0) = H1(X, X0), Ext

2OX0 (X0 , OX0) = H

2(X, X0) = 0.

We now return to the original bordered Riemann surface .

Definition 3.1. LetO be the sheaf of local holomorphic functions on with realboundary values.

Let be a sheaf of O-modules, together with anR derivation d : O ,which satisfy the following universal property: for any sheaf ofO-modules F, and

for anyR derivationd : O F, there exists a unique O-module homomorphismf : Fsuch that d = f d. We call the sheaf of Kahler differentials on .

Let = HomO(, O) = , the dual of in the category of sheaves ofO-modules.Note that and are locally free sheaves ofO-modules of rank 1. Analyt-

ically, is the sheaf of local holomorphic 1-forms on whose restriction to are real 1-forms, and is the sheaf of holomorphic vector fields with boundaryvalues in T. There are natural isomorphisms

ExtiO(1, O) = ExtiOX0 (

1X0 , OX0), Hi(, ) = Hi(X0, X0)


10/85

10

for i 0. Since the first order deformations of the bordered Riemann surface ,of the symmetric Riemann surface (C, ), and of the real algebraic curve X0 areidentified, the first order deformation of is canonically identified with

Ext1O(, O) = H1(, ),and the obstruction lies in

Ext2O(, O) = H2(, ) = 0.3.1.2. Analytic approach. We first give the definitions of a differentiable family ofcompact symmetric Riemann surfaces and a differentiable family of bordered Rie-mann surfaces, which are modifications of [Ko, Definition 4.1].

Definition 3.2. Suppose given a compact symmetric Riemann surface (Mt, t) foreach point t of a domain B ofRm. {(Mt, t)|t B} is called a differentiable familyof symmetric Riemann surfaces if there are a differentiable manifoldM, a surjectiveC map :

M B and a C map :

M Msuch that

(1) The rank of the Jacobian matrix of is equal to m at every point of M.(2) For each t B, 1(t) is a compact connected subset of M.(3) 1(t) = Mt.(4) There are a locally finite open covering{Ui | i I} ofM andC functions

zi :Ui C such that{Ui 1(t), zi|Ui1(t) | i I,Ui 1(t) = }

is an analytic atlas for Mt.(5) = , and |1(t) = t : Mt Mt is an antiholomorphic involution.

Definition 3.3. Suppose given a bordered Riemann surface Mt for each point tof a domain B ofRm. {Mt | t B} is called a differentiable family of borderedRiemann surfaces if there are a differentiable manifold with boundary

Mand a

surjective C map : M B such that(1) The rank of the Jacobian matrix of is equal to m at every point of M.(2) For each t B, 1(t) is a compact connected subset of M.(3) 1(t) = Mt.(4) There are a locally finite open covering{Ui | i I} ofM andC functions

zi :Ui C+ such that{Ui 1(t), zi|Ui1(t) | i I,Ui 1(t) = }

is an analytic atlas for Mt.

The complex double C of a bordered Riemann surface is a complex manifoldof dimension 1. Infinitesimal deformation of C can be identified with

H

1

(C

, TC) = H

1

(X, X),and the obstruction lies in

H2(C, TC) = H2(X, X) = 0.

(See [Ko].) The differential d of is an antiholomorphic involution on the holo-morphic line bundle TC C which covers : C C. d induces a complexconjugation

: H1(C, TC) H1(C, TC)


11/85

11

which gets identified with the action of S on Ext1OX (X , OX) under the isomor-phism

H1

(C, TC) = Ext1

OX (X , OX).The pair (TC , d) is the holomorphic complex double of the Riemann-Hilbert

bundle (T, T) (, ), where a Riemann-Hilbert bundle and its holomorphiccomplex double are defined in [KL, Section 3.3.4]. There is an isomorphism (see[KL, Section 3.4])

H1(C, TC) = H1(, , T, T).

From the above discussion, we know that H1(C, TC) gives first order deformation

of the symmetric Riemann surfaces (C, ). Given a differential family of symmetric

Riemann surface (t, t) such that (0, 0) = (C, ), Kt = t/t is a family ofKlein surfaces. Each Kt is homeomorphic to , which is orientable, so it admits twoanalytic structures compatible with its dianalytic structure, and one is the complexconjugate of the other. We get two differentiable families Mt, Mt = Mt of bordered

Riemann surfaces, one is a deformation of , and the other is a deformation of .So H1(, , T, T) should give infinitesimal deformations of .

Recall that an infinitesimal deformation of C determines a Cech 1 cocycle inH1(A, C) H1(C, TC), where A is an analytic atlas of C, and C is the sheafof local holomorphic vector fields on C [Ko]. (The inclusion is an isomorphism

if A is acyclic). Following argument similar to that in [Ko], we now show that aninfinitesimal deformation of determines a Cech 1 cocycle in H1(A, ), where Ais an analytic atlas of , and is the sheaf of local holomorphic vector fields onT with boundary values in T.

Let {Mt | t B} be a differentiable family of bordered Riemann surfaces,M0 = . We use the notation in Definition 3.3. Then

A = {(Ui, i) Ui 1(0), zi|Ui1(t) | i I,Ui

1(t) = }is an analytic atlas of . Without loss of generality, we may assume that A is acyclic.We define t-dependent transition functions fij by zi = fij(zj , t) = fik(zk, t). Thenfij(zj , t) R if zj R by 4. of Definition 3.3.

zi = fik(zk, t) = fij(fjk(zk, t), t)

fikt

=fijzj

fjkt

+fij

t

Multiplying by zi and noting thatfijzj

= zizj , we have

fikt

zi=

fjkt

zj+

fijt

zi

Let (xi, yi) be real coordinates defined by zi = xi + iyi, then the boundary isdefined by {yi = 0}, and the tangent line to the boundary is spanned by xi .Under the isomorphism T T0,1 given by v (v iJ v)/2, we have x zand y i z . So ik fikt

t=0

zi

defines a Cech 1 cochain in C1(A, ) whichsatisfies the cocycle condition ik = ij + jk .

By exactly the same argument in [Ko] we see that another system of coordinateswill give rise to a Cech 1 cocycle = +, where is a Cech 0 cochain. Therefore,


12/85

12

1

0

0

0

1 1

01

00

2

Figure 1. strata of M2

the infinitesimal deformation of is given by

H1(A, ) = H1(, , T, T).3.2. Nodal bordered Riemann surfaces. To compactify the moduli of bordered

Riemann surface, we will allow nodal singularities. The complex double of a bor-dered Riemann surface is a complex algebraic curve with real structure. The stablecompactification of moduli of such curves parametrizes stable complex algebraiccurves with real structure [SS, Se], or equivalently, stable compact symmetric Rie-mann surfaces. Naively, the quotient of a stable compact symmetric Riemann sur-face by its antiholomorphic involution will give rise to a stable bordered Riemannsurface. We will make this idea precise in this section.

Let be a (smooth) bordered Riemann surface of type (g, h). Note that if : is an automorphism (Definition 2.14), then its complex double ([KL,Section 3.3.2]) C : C C is an automorphism of (C, ) (Definition 2.7). Thisgives an inclusion Aut() Aut(C, ). It is easy to see that the following areequivalent:

is stable, i.e., Aut() is finite. C is stable. The genus g = 2g + h 1 of C is greater than one. The Euler characteristic () = 2 2g h of is negative.

We start with g = 2. Let M2 be the moduli of stable complex algebraic curvesof genus 2. The strata of M2 are shown in Figure 1.

The dual graph and the underlying topological surface of a prestable curve areshown. In the dual graph of the curve C, each vertex corresponds to an irreducible


13/85

13

1

1

3

3

3

3 63 6 3

61 3 3

Figure 2. strata of M(0,3). Each stable bordered Riemann sur-face above represents a topological type, and the number aboveit is the number of strata associated to it. Strata associatedto the same topological type are related by relabelling the three

boundary circles. There are one 3-dimensional stratum, nine 2-dimensional strata, twenty-one 1-dimensional strata, and fourteen0-dimensional strata. We will see in Example 4.7 that M(0,3) canbe identified with the associahedron K5 defined by J. Stasheff[St].

component of C, labeled by the genus of the normalization, while each edge cor-responds to a node of C, whose two end points correspond to the two irreduciblecomponents which intersect at this node.

If g = 2, (g, h) can be (0, 3) (Figure 2) or (1, 1) (Figure 3).

Definition 3.4. Let(x, y) be coordinates onC2, andA(x, y) = (x, y) be the complexconjugation. A node on a bordered Riemann surface is a singularity isomorphic toone of the following:

(1) (0, 0) {xy = 0} (interior node)(2) (0, 0) {x2 + y2 = 0}/A (boundary node of type E)(3) (0, 0) {x2 y2 = 0}/A (boundary node of type H)

A nodal bordered Riemann surface is a singular bordered Riemann surface whosesingularities are nodes.

A type E boundary node on a bordered Riemann surface corresponds to a bound-ary component shrinking to a point, while a type H boundary node corresponds


14/85

14

Figure 3. strata of M(1,1)

to a boundary component intersecting itself or another boundary component. Theboundary of a nodal Riemann surface is a union of points and circles, where onecircle might intersect other circles in finitely many points.

The notion of morphisms and complex doubles can be easily extended to nodalbordered Riemann surfaces. The complex double of a nodal bordered Riemannsurface is a nodal compact symmetric Riemann surface.

Example 3.5. Consider C = {X2 Y2 + Z2 = 0} P2, where [X, Y, Z] arehomogeneous coordinates on P2, and R. C is invariant under the standardcomplex conjugation A([X , Y , Z ]) = [X, Y , Z] on P2, so = A|C is an antiholo-morphic involution on C. For = 0, (C, ) is a symmetric Riemann surface oftype (0, 1, 0), and C/ is the disc, which is a bordered Riemann surface. C0 hastwo irreducible components {X + Y = 0} and {X Y = 0} which are projectivelines, and the intersection point [0, 0, 1] of the two lines is a node on C0. Both linesare invariant under the antiholomorphic involution 0. C0/0 is a nodal borderedRiemann surface: it is the union of two discs whose intersection is a boundary nodeof type H.

Example 3.6. Consider C =

{X2 + Y2 + Z2 = 0

} P2, where

R. C is

invariant under the complex conjugation A on P2, so = A|C is an antiholo-morphic involution on C. Set = C/. For > 0, (C, ) is a symmetricRiemann surface of type (0, 0, 1), and is the real projective plane; for < 0,(C, ) is a symmetric Riemann surface of type (0, 1, 0), and is the disc. C0has two irreducible components {X + 1Y = 0} and {X 1Y = 0} whichare projective lines, and their intersection point [0, 0, 1] is a node on C0. The an-tiholomorphic involution 0 interchanges the two irreducible components ofC0 andleaves the node invariant, so 0 = P1, which is a smooth Riemann surface without


15/85

15

normalizationdegeneration

(g, h) = (1, 3)

(g, h) = (1, 2)

Figure 4. boundary node of type E

degeneration normalization

(g, h) = (1, 3)

(g, h) = (1, 2)

Figure 5. boundary node of type H1

boundary. However, we would like to view it as a disc whose boundary shrinks to apoint which is a boundary node of type E.

Definition 3.7. Let be a nodal bordered Riemann surface. The antiholomorphic

involution on its complex double C can be lifted to : C C, where C isthe normalization of C (viewed as a complex algebraic curve). The normalization

of = C/ is defined to be = C/.From the above definition, the complex double of the normalization is the nor-

malization of the complex double, i.e., C = C.Let be a smooth bordered Riemann surfaces of type ( g, h). The following are

possible degenerations of whose only singularity is a boundary node.

E. One boundary component shrinks to a point. The normalization is a smoothbordered Riemann surface of type (g, h 1) (Figure 4).

H1. Two boundary components intersect at one point. The normalization is a

smooth bordered Riemann surface of type (g, h1), and the two preimagesof the node are on the same boundary component (Figure 5).

H2. One boundary component intersects itself, and the normalization of thesurface is connected. The normalization is a smooth bordered Riemannsurface of type (g 1, h + 1), and the two preimages of the node are ondifferent boundary components (Figure 6).

H3. One boundary component intersects itself, and the normalization of thesurface is disconnected. The normalization is a disjoint union of two smooth


16/85

16


(g, h) = (1, 3) (g, h) = (0, 4)



(g, h) = (1, 3) (g1, h1) = (1, 2) (g2, h2) = (0, 2)


bordered Riemann surfaces of types (g1, h1) and (g2, h2) such that g =g1 + g2 and h = h1 + h2 1, and each connected component contains oneof the two preimages of the node (Figure 7).

Definition 3.8.A prestable bordered Riemann surface is either a smooth borderedRiemann surface or a nodal bordered Riemann surface.

Let be a prestable bordered Riemann surface, be its normalization. LetC1, . . . , C , 1, . . . , be the connected components of , where Ci is a smoothRiemann surface of genus gi, and i is a smooth bordered Riemann surface oftype (gi , hi). Let be the number of connecting interior nodes (Figure 8), andE , H1, H2, H3 be the numbers of boundary nodes described in E, H1, H2, H3,respectively.

The topological type (g, h) of is given by

g = g1 + + g + g1 + + g + + H2h = h1 + + h + E + H1 H2 H3

It is now straightforward to extend the notion of marked bordered Riemannsurfaces to prestable bordered Riemann surfaces.

Definition 3.9. Leth be a positive integer, g, n be nonnegative integers, and m =(m1, . . . , mh) be an h-uple of nonnegative integers. A prestable marked borderedRiemann surface of type (g, h) with (n, m) marked points is an (h + 3)-uple

(, B; p; q1, . . . , qh)

whose components are described as follows.


17/85

17

connecting interior node disconnecting interior node

(g, h) = (2, 1)

Figure 8. connecting and disconnecting nodes

is a prestable bordered Riemann surface of type (g, h).

B = (B1, . . . , Bh), where = hi=1 Bi, and eachBi is an immersed circle.The circles B1, . . . , Bh may intersect with each other at boundary nodes,and become h disjoint embedded circles under smoothing of all boundarynodes.

p = (p1, . . . , pn) is an n-uple of distinct smooth points in. qi = (qi1, . . . , qimi) is an mi-uple of distinct smooth points onBi.

Definition 3.10. A morphism between prestable marked bordered Riemann sur-faces of type (g, h) with (n, m) marked points

(, B; p; q1, . . . , qh) (, B; p; (q)1, . . . , (q)h)is an isomorphism of prestable bordered Riemann surfaces : such that(Bi) = (B)i for i = 1, . . . , h, (pj) = pj for j = 1, . . . , n, and (q

ik) = (q

)ik fork = 1, . . . , mi. A prestable marked bordered Riemann surfaces of type (g, h) with

(n, m) marked points is stable if its automorphism group is finite.

Remark 3.11. The category of prestable marked bordered Riemann surfaces oftype (g, h) with (n, m) marked points is a groupoid since every morphism in Defi-nition 3.10 is an isomorphism.

Example 3.12. Consider the case (g, h) = (0, 2), n = 0, m = (1, 0) (annuli withone boundary marked point). The moduli space M(0,2)(0,(1,0)) is an interval [0, 1].There are three strata: t (0, 1), t = 0, t = 1 (Figure 9)Example 3.13. Consider the case (g, h) = (0, 2), n = 0, m = (2, 0) (annuli withtwo boundary marked points on the same boundary circle). The moduli spaceM(0,2)(0,(2,0)) is a pentagon. There are eleven strata (Figure 10).

Example 3.14. Consider the case (g, h) = (0, 2), n = 0, m = (1, 1) (annuli withone marked point on each boundary circle). The moduli space M(0,2)(0,(1,1)) is adisc {z C | |z| 1}. There are four strata: 0 < |z| < 1, |z| = 1 but z = 1, z = 1,z = 0 (Figure 11)

3.3. Deformation theory for prestable bordered Riemann surfaces. Thealgebraic approach of deformation theory for smooth bordered Riemann surfacesin Section 3.1 can be easily extended to nodal bordered Riemann surfaces. We willalso consider marked points.


18/85

18

1

1 1

t = 0t = 1

t (0, 1)

Figure 9. strata of M(0,2)(0,(1,0))

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1 2

1

2

1

2

1

2

Figure 10. strata of

M(0,2)(0,(2,0))

Let (, B; p; q1, . . . , qh) be a marked prestable bordered Riemann surface of type(g, h) with (n, m) marked points. We want to study its infinitesimal deformation.The ordering of the boundary circles is irrelevant to the infinitesimal deformation,so in this section we will ignore it and write (; p; q), where

q = (q1, . . . , qm) = (q11 , . . . , q

1m1 , q

21 , . . . , q

2m2 , . . . , q

h1 , . . . , q

hmh),


19/85

19

0 < |z| < 1

|z| = 1, z = 1

z = 0 z = 1

Figure 11. strata of M(0,2)(0,(1,1))

and m = m1 + + mh.The complex double (C, ) of is a nodal complex algebraic curve X which

is the complexification of some real algebraic curve X0, i.e., X = X0 R C. Al-ternatively, (X, S) is a complex algebraic curve with a real structure, where S is asemi-linear automorphism which induces the antiholomorphic involution on C.

Algebraically, the complex double of (; p; q) is a nodal complex algebraic curvewith (2n + m) marked points (X, x), where

x = (x1, . . . , x2n+m) = (p1, . . . , pn, p1, . . . , pn, q1, . . . , qm).

Here we identify with the image under the inclusion i : C, and denote (p)by p.

Let

Dx = x1 + + x2n+mbe the divisor in X associated to x. The set of first order deformation of thepointed complex algebraic curve (X, x) is canonically identified with the complexvector space

Ext1OX (X(Dx), OX),and the obstruction lies in

Ext2OX (X(Dx), OX).

We claim that Ext2OX (X(Dx), OX) = 0. Three terms in the local to globalspectrum sequence contribute to Ext2OX (X(Dx), OX):

H0(X, Ext2OX (X(Dx), OX)),H1(X, Ext1OX (X(Dx), OX)),H2(X, Ext0OX (X(Dx), OX)).


20/85

20

The curve X has only nodal singularities, so

Ext2

OX

(X(Dx),

OX) = 0,

thus

H0(X, Ext2OX (X(Dx), OX)) = 0.The sheaf Ext1OX (X(Dx), OX) is supported on nodes, so

H1(X, Ext1OX (X(Dx), OX)) = 0.Finally,

H2(X, Ext0OX (X(Dx), OX)) = 0since X is one dimensional. So we get the desired vanishing.

The semi-linear automorphism S induces a complex conjugation

S : Ext1OX (X(Dx), OX) Ext1OX (X(Dx), OX).

The fixed locus Ext1OX (X(Dx), OX)S gives the first order deformation of (X, x, S)

as a pointed complex algebraic curve with a real structure.We will study the group Ext1OX (X(Dx), OX) and the action of S on it more

closely. We first introduce some notation.Let C1,...,C be the irreducible components of which are (possibly nodal)

Riemann surfaces, and let 1,..., be the remaining irreducible components of, which are (possibly nodal) bordered Riemann surfaces. Then the irreduciblecomponents of X are

C1, . . . , C , C1, . . . , C , (1)C, . . . , ()C.

Let Ci denote the normalization of Ci, i = 1, . . . , , and let i denote thenormalization of i , i = 1, . . . , . Then

C1, . . . ,

C ,

1, . . . ,

are the connected components of , the normalization of , and

C1, . . . , C ,C1, . . . ,

C , (1)C, . . . , ()C.

are the connected components of X, the normalization of X.Let r1, . . . , rl0 be interior nodes of , and s1, . . . , sl1 be boundary

nodes of . Then X has 2l0 + l1 nodes r1, . . . , rl0 , r1, . . . , rl0 , s1, . . . , sl1 .

Let pj be the preimage of pj under the normalization map : X X, j =1, . . . , n. Define pj , qj similarly. Let r, rl0+ be the preimages ofr, = 1, . . . , l0,and define s , sl1+ similarly.

Consider X with marked points

x = (p1, . . . , pn, p1, . . . , pn, q1, . . . , qm, r1, . . . , r2l0 , r1, . . . , r2l0 , s1, . . . , s2l1),

which can be written as a disjoint union of pointed curves

(C, yi), ( C, yi), ((i )C, zi) = ((i )C, p

i , pi, qi

),

where

yi = (yi1, . . . , yini), y

i = (yi1, . . . , yini),

pi

= (pi

1 , . . . , pi

ni), pi

= (pi

1 , . . . , pi

ni), qi

= (qi

1 , . . . , qi

mi),


21/85

21

i = 1, . . . , , i = 1, . . . , , pi

1 , . . . , pini

i , qi

1 , . . . , qimi

i , andi=1

ni +

i=1

ni = n + 2l0,

i=1

mi = m + 2l1.

From the local to global spectrum sequence, we have an exact sequence

0 H1(X, Ext0OX (X(Dx), OX)) Ext1OX (X(Dx), OX) H0(X, Ext1OX (X(Dx), OX)) 0,

where

Ext0OX (X(Dx), OX) = HomOX (X(Dx), OX) = 1X(Dx).We have the following elementary fact:

Lemma 3.15.

1X(Dx) = TX(Dx),

where : X X is the normalization map, and Dx is the divisor correspondingto the marked points x in X.

Proof. The equality obviously holds for smooth points. It suffices to show that1Y = TY for Y = SpecC[x, y]/(xy), which follows from a local calculation. 2

The map : X X is an affine morphism, so by [Ha, Chapter III, Exercise 4.1]we have

H1(X, Ext0OX (X(Dx), OX)) = H1(X, TX(Dx)) =i=1

(Wi Wi) i=1

Wi

where the vector spaces

Wi = H

1

(

Ci, TCi(Dyi

)),

Wi = H

1

(

Ci, TCi(Dyi

)),Wi = H

1((i)C, T(i)C(Dzi )

correspond to deformations of pointed curves (Ci, yi), (Ci, yi), ((i )C, zi

), respec-

tively.Another local calculation shows that

H0(X, Ext1OX (X(Dx), OX)) =l0=1

(V V) l1

=1

V ,

where

V = TrX Trl0+X, V = TrX Trl0+X, V = Ts X Tsl0+ X

correspond to smoothing of the nodes r, r, s in X, respectively.Now the action of S on Ext1OX (X(Dx), OX) is clear: it acts on Wi Wi and

V V by (a, b) (b, a), and it acts on Wi and V by a a. The real vectorspace Wi

Scorresponds to deformation of the pointed symmetric Riemann surface

((i)C, zi, ). We also have Wi

S = Wi , whereWi = H

1(i , i , Ti (pi

1 pi

ni), Ti (q

i

1 . . . qi

mi))

corresponds to deformation of the pointed bordered Riemann surface (i , pi , qi

).


22/85

22

crosscap

t>0 t=0 t0 t 0 to < 0 (Figure

12, 13, 14, 15). C corresponds to V , R corresponds to VS = V , and 0 corresponds to V+ .

Above discussion can be summarized as follows.

(1) The infinitesimal deformation of the pointed complex algebraic curve (X, x)is given by

Ext1OX (X(Dx), OX) =i=1

(Wi Wi) i=1

Wi l0=1

(V V) l1

=1

V .

(2) The infinitesimal deformation of the pointed complex algebraic curve witha real structure (X, x, S) is given by

Ext1OX (X(Dx), OX)S =i=1

(Wi Wi)S i=1

WiS

l0=1

(V V)S l1

=1

VS

.


23/85

23

t>0 t=0 t0 t=0 t 1, and showedthat the moduli space is compact and Hausdorff in the topology. In this section, wedescribe how the above works can be modified to study stable bordered Riemannsurfaces.

4.1. Various moduli spaces and their relationships. Let

(, B; p; q1, . . . , qh)


24/85

24

be a marked stable bordered Riemann surface of type ( g, h) with (n, m) markedpoints. The moduli of the ordering of the circles B1, . . . , Bh is given by the permu-tation group of h elements. Let (C, ) be the complex double of , and let

x = (x1, . . . , x2n+m) = (p1, . . . , pn, p1, . . . , pn, q1, . . . , qm)

(C, , p1, x) is a stable symmetric Riemann surface of genus g = 2g + h 1 with(n, m) marked points. Removing x1, . . . , xn from C, where n = 2n + m, we obtain(S, ), a stable symmetric Riemann surface of genus g with n punctures. Let S

be the complement of nodes in S. There is a one to one correspondence betweenconnected components of S and irreducible components of S. Each connectedcomponents of S is a smooth punctured Riemann surface. The stability conditionis equivalent to the statement that each connected component of S has negativeEuler characteristic. Therefore, there is a unique complete hyperbolic metric in theconformal class of Riemannian metrics on S determined by the complex structure.

Let Mg,n be the moduli of stable compact Riemann surfaces of genus g with nmarked points, or equivalently, the moduli of stable complex algebraic curves ofgenus g with n marked points. Let Pg,n be the moduli of stable oriented hyperbolicsurfaces of genus g with n punctures. From the above discussion we know thatthere is a surjective map : Mg,n Pg,n. is generically n! to one since themarked points are ordered, while the punctures are not. The fiber over a point inPg,n represented by the surface S consists of less than n! points if and only if thereis an automorphism of S permuting its punctures.

Let MRg,(n,m) be the moduli of stable symmetric compact Riemann surface of

genus g with (n, m) points, and let MR(g,h,k),(n,m) be the moduli of smooth symmetric

compact Riemann surface of type (g , h, k) with (n, m) marked points. Note thatMR(g,0,k),(n,m) is empty ifm > 0. M

R

(g,h,k),(n,m) are disjoint subsets ofMR

g,(n,m), and

their closures MR

(g,h,k),(n,m) (in the topology defined later) cover MR

g,(n,m).There is an involution A : Mg,n Mg,n, given by

[(, x1, . . . , xn)] [(, (xn+1), . . . , (x2n), (x1), . . . , (xn), (x2n+1), . . . , (x2n+m))]

where : is the canonical anti-holomorphic map from to its complexconjugate . Let MAg,n denote the fixed locus of A. Then there is a surjective map

MRg,(n,m) MAg,n, given by forgetting the symmetry . This map is genericallyinjective. It fails to be injective exactly when the automorphism group of ( , x) islarger than that of (, , x).

Let PRg,n be the moduli of stable symmetric oriented hyperbolic surfaces of genus

g with n punctures, and let PR(g,h,k),n the moduli of smooth symmetric oriented

hyperbolic surfaces of type (g , h, k) with n punctures. PR(g,h,k),n are disjoint subsets

ofPRg,n, and their closures PR

(g,h,k),n (in the topology to be defined later) cover PRg,n.

There is an involution A : Pg,n Pg,n, given by [S] [S]. There is a surjectivemap PRg,n PA

g,n, given by forgetting the symmetry . This map is generically

injective. It fails to be injective exactly when the automorphism group of S islarger than that of (S, ).

We have the following commutative diagrams:


25/85

25

Mg,nA Mg,n

Pg,n

A Pg,n

MRg,(n,m) MAg,nR A

PRg,(n,m) PA

g,n

where the generic fiber of R consists of 2nn!m! points. The factor 2n correspondsto the permutation of the two points in each of the n conjugate pairs. The factor n!corresponds to the permutation of the n conjugate pairs. The factor m correspondsto the permutation of the m marked points fixed by the symmetry. Similarly,the generic fiber of A consists of 2nn!m! points. For a generic point in PA

g,n, its

preimage under consists of n! points, but only 2nn!m! lie in the fixed locus of A.Neither R nor A is surjective because the number of punctures fixed by the

symmetry can be any integer between 0 and n, not only m.We are interested in the moduli space M(g,h),(n,m) of stable bordered Riemann

surfaces of type (g, h) with (n, m) marked points. There is a finite to one mapM(g,h),(n,m) MRg,(n,m) given by complex double, so there is a finite to one mapM(g,h),(n,m) PR(g,h,0),n PRg,n. We will first study Pg,n and PRg,n, following[Ab, Se].

4.2. Decomposition into pairs of pants. A pair of pants P is a sphere fromwhich three disjoint closed discs (or points) have been removed. It is the interiora stable bordered Riemann surface of type (0, 3). There is a unique hyperbolicstructure compatible with the complex structure of P such that the boundarycurves are geodesics. Conversely, given a hyperbolic structure on P such thatthe boundary curves are geodesics, the conformal structure is determined up toconformal or anti-conformal equivalence by the lengths l1, l2, and l3 of the threeboundary curves ([Ab, Chapter II (3.1), Theorem]).

4.2.1. Riemann surfaces with punctures. A Riemann surface S of genus g with npunctures can be decomposed into pairs of pants. More precisely, there are 3g3+ndisjoint curves 1, . . . , 3g3+n on , each of which is either a closed geodesic (inthe hyperbolic metric) or a node, such that the complement of 3g3+ni=1 i is adisjoint union of 2g 2 + n pairs of pants P1, . . . , P 2g2+n. A boundary componentof the closure of a pair of pants in this decomposition is either a decomposing curveor a puncture. We call

P= {P1, P2, . . . , P 2g2+n}a geodesic decomposition of S into pairs of pants.

Suppose that there exists an anti-holomorphic involution : S S. Then (S, )is a stable symmetric Riemann surface of genus g with n punctures, and is anisometry of the hyperbolic metric. Let

P= {P1, P2, . . . , P 2g2+n}be a geodesic decomposition of S into pairs of pants. Then

(P) = {(P1), (P2), . . . , (P2g2+n)}is another geodesic decomposition of S into pairs of pants. P is said to be -invariant if (P) = P. The argument in [Se, Section 4], combined with [Ab,Chapter II (3.3), Lemma 3], shows that


26/85

26

Theorem 4.1. Let(S, ) be a stable symmetric Riemann surface of genus g withn punctures. There exists a -invariant geodesic decomposition of into pairs ofpants such that the decomposing curves are simple closed geodesics of length lessthan C(g, n), where C(g, n) is a constant depending only on g, n.

4.2.2. Riemann surfaces with boundary and punctures. Let (, B; p) be a stable

bordered Riemann surface of type (g, h) with (n,0) marked points, and supposethat has no boundary nodes. Let S be the complement of marked points in ,then S is a surface of type (g, h, n) in the sense of [Ab], namely, S is obtainedby removing h open discs and n points from a compact (possibly nodal) Riemannsurface of genus g, where the discs and points are all disjoint. S is stable in thesense that its automorphism group is finite. Let S be the complement of nodes inS. Each connected component of S is a smooth Riemann surface with boundaryor punctures. The stability condition on S is equivalent to the statement that eachconnected component of S has a negative Euler characteristic, so there exists aunique hyperbolic metric on S in the conformal class determined by the complexstructure such that the boundary circles are geodesics.

Let S be a stable surface of type (g, h, n). The S can be decomposed into pairsof pants. More precisely, there are 3g + h 3 + n disjoint curves 1, . . . , 3g+h3+non S, each of which is either a closed geodesic (in the hyperbolic metric) or a node,

such that the complement of 3g+h3+ni=1 i is a disjoint union of 2g + h 2 + npairs of pants P1, . . . , P 2g+h2+n. A boundary component of the closure of a pairsof pants in this decomposition is a decomposing curve, a boundary component ora puncture. We call

P= {P1, P2, . . . , P 2g+h2+n}a geodesic decomposition of S into pairs of pants. We have the following result([Ab, Chapter II (3.3), Lemma 3]):

Theorem 4.2. LetS be a stable surface of type (g, h, n). There is a geodesic decom-position ofS into pairs of pants such that the decomposition curves are simple closedgeodesic with length less than C(g,h,n,L1, . . . , Lh), where C(g,h,n,L1, . . . , Lh)is a constant depending only on g, h, n, and the lengths of the h border curvesL1, . . . , Lh.

We will see later that the moduli of stable surfaces of type (g, h, n) is of (real)dimension 6g + 3h 6 + 2n, and L1, . . . , Lh are among the 6g + 3h 6 + 2n realparameters. They are not good coordinates for compactness since L1, . . . , Lh canbe arbitrarily large. Actually, the length of some border curve tends to infinity asS acquires a type H boundary node. To deal with boundary nodes and boundarymarked points, we go to the complex double, where the local coordinates of themoduli can be chosen to be bounded.

4.3. Fenchel-Nielsen coordinates.

Definition 4.3. Let S be a stable symmetric Riemann surface of genus g with npunctures. A geodesic pants decomposition Pis oriented if

(1) The pairs of pants in Pis ordered.(2) The boundary components of each pair of pants in P is ordered.(3) Any decomposing curve which is not a node is oriented.


27/85

27

1

2 31

2

32,3

3,11,2

Figure 16. a pair of pants

Remark 4.4. The orientability of a geodesic decomposition of a surface of type(g, h, n) can be defined similarly, with the additional assumption that the boundary

components are ordered.

Let P be a pair of pants with hyperbolic structure with ordered boundary com-ponents 1, 2, 3. The base points i on i, i = 1, 2, 3, are shown in Figure 16.In Figure 16, 1,2 is the geodesic which realizes the distance between 1, 2, etc.

IfPis oriented, then each boundary curve of a pair of pants in Phas a base point.Each decomposing curve has two distinguished points since it appears twice as aboundary of some pair of pants in P. The orderings in 1. and 2. of Definition 4.3determine an ordering of the decomposing curves j , j = 1, . . . , 3g 3 + n andan ordering on the two distinguished points 1j ,

2j on each decomposing curve. We

define lj(P) to be the length ofj , and j(P) be the distance one travels from 1j to2j along j , in the direction determined by 3. in Definition 4.3. Set j(P) = 2 jljif lj

= 0. We have lj

0, 0

j < lj , and 0

j < 2.

Definition 4.5. LetS, S be two stable Riemann surfaces of genus g with n punc-tures. Let S, S be 1-dimensional complex manifolds obtained from S, S by re-moving the nodes. A strong deformation : S S is a continuous map suchthat

(1) If r is a node on S, then (r) is a node on S.(2) If r is a node on S, then 1(r) is a node or an embedded circle on a

connected component of S.(3) f|1(S) : 1(S) S is a diffeomorphism.

Note that 1. implies that 1(S) S, so 3. makes sense. There is a strongdeformation : S S if and only if S can be obtained by deforming S as aquasiprojective variety over C.

Remark 4.6. A strong deformation : (S, ) (S, ) between stable symmetricRiemann surfaces can be defined similarly, with the additional assumption that = . A strong deformation between two surfaces of type (g, h, n) can alsobe defined similarly.

We now describe Fenchel-Nielsen coordinates for various category of surfaces.

(1) Let S be a stable Riemann surface of genus g with n punctures. Let Pbe an oriented geodesic decomposition of S into pairs of pants, and let


28/85

28

1, 2, . . . , 3g3+n be the decomposing curves ofP. Suppose that there isa strong deformation : S S. Let j be the closed geodesic homotopic to

1

(j). There exists another strong deformation such that (j) = j ,so Pis pulled back under to an oriented geodesic decomposition PS .

[S] (l1(PS), 1(PS), . . . , l3g3+n(PS), 3g3+n(PS))defines local coordinates (l1, 1, . . . , l3g3+n, 3g3+n) on Pg,n. Therefore,both Mg,n and Pg,n are (6g 6 + 2n) dimensional.

(2) Let (S, ) be a stable symmetric surface of genus g with n punctures,and let P be an oriented geodesic decomposition of S into pairs of pantswhich is invariant under . By considering -invariant decompositionsinto pants in a neighborhood of (S, ) in PRg,n we obtain local coordinates(l1, 1, . . . , l3g3+n, 3g3+n). However, these parameters are not indepen-dent. If(i) = j , i = j, then li = lj , and i = c j for some constant c.If(i) = i, then i = 0. So there are 3g 3 + n independent parameters,and the dimension ofPRg,n is 3g 3 + n.

(3) Let S be a stable surface of type (g, h, n), and let R1, . . . , Rh be its bordercurves. Let P be an oriented geodesic decomposition of S into pairs ofpants, and let 1, . . . , 3g+h3+n be the decomposing curves. We havelocal coordinates (l1, 1, . . . , l3g+h3+n, 3g+h3+n, L1, . . . , Lh), where lj isthe length of j , j is the angle of gluing along j , and Li is the lengthof the border curve Ri. Therefore, the moduli of stable surfaces of type(g, h, n) is 6g + 3h 6 + 2n = 3g 3 + n, where g = 2g + h 1, and n = 2n.This is consistent with the previous paragraph since the complex double ofS is a stable symmetric Riemann surface with genus g = 2g + h 1 andn = 2n punctures.

(4) Let (, B; p; q1, . . . , qh) be a stable marked bordered Riemann surface of

type (g, h) with (n, m) marked points. (C, , x) be its complex double, and(S, ) be the symmetric Riemann surface obtained from by removing themarked points, as in Section 4.1. Choose a -invariant geodesic decomposi-tion ofS into pants, we have local coordinates (l1, 1, . . . , l3g3+n, 3g3+n),as described in 1., where

g = 2g + h 1, n = 2n + m1 + + mh,3g 3 + n = 6g + 3h 3 + 2n + m1 + + mh.

We have seen that half of the 2(6g + 3h6+ 2n+ m1+ + mh) parametersare independent, so the dimension of M(g,h),(n,m) is

6g + 3h 6 + 2n + m1 + + mh.In the following example, we describe the Frenchel-Nelson coordintates of the

moduli space M0,3 of a pair of pants explicitly.

Example 4.7. The hexigon in Figure 17 is obtained by cutting the pair of pantsin Figure 16 along the geodesics 1,2, 2,3, 3,1. 1 is the geodesic which realizes thedistance between 1 and 2,3, etc.

Letl1, l2, l3, l4, l5, l6, l7, l8, l9 be twice the lengths of1, 2, 3, 2,3, 3,1, 1,2, 1, 2, 3,respectively. The degenerationli = 0 corresponds to a real codimension one stratumVi of M0,3. Let Vij = Vi Vj and Vijk = Vi Vj Vk. M0,3 can be identified with


29/85

29

1

2 3

1,2

2,3

3,11

23

V12

V13

V23

V45

V56

V46

V14

V25

V36

V27

V37

V57

V67

V18

V38

V48

V68

V19

V29

V49

V59

Figure 17.

the associahedron K5 defined by J. Stasheff [St]. The configuration of the strata inM0,3 = K5 is shown in Figure 17. There is one 3-dimensional stratum. There arenine 2-dimensional strata:

V1, V2, V3, V4, V5, V6, V7, V8, V9.

There are twenty-one 1-dimensional strata:

V12, V13, V23, V45, V56, V46, V14, V25, V36, V27, V37, V57, V67, V18, V38, V48, V68, V19, V29, V49, V59.

There are fourteen 0-dimensional strata:V123, V456, V237, V257, V367, V567V138, V148, V368, V468V129, V149, V259, V459.

There is a one-to-one correspondence between the 0-dimensional strata and Frenchel-Nelson coordinate charts of M0,3: the Frenchel-Nelson coordinates near Vijk areli, lj , lk.

4.4. Compactness and Hausdorffness. We first define a topology on M(g,h),(n,m),following [Ab, Chapter II (3.4)], and [Se, Section 5]. We will call it Fenchel-Nielsentopology.

Definition 4.8. A strong deformation between two stable marked bordered Rie-mann surfaces (, B; p; q1, . . . , qh) and(, B; p; q1, . . . , qh) of type (g, h) with(n, m)

marked points is a continuous map : such that

(1) (Bi) = Bi, (qik) = qik, (pj) = pj.

(2) If r is an interior node on , then (r) is an interior node on .

(3) If s is a boundary node of type E (H) on , then (s) is a boundary nodeof type E (H) on .

(4) If r is an interior node on , then 1(r) is an interior node or a circle.(5) If s is a boundary node of type E, then 1(r) is a boundary node of type

E or a border circle.


30/85

30

(6) If s is a boundary node of type H, then 1(s) is a boundary node or typeH or an arc with ends in .

(7) f|

1

(S):

1(S

)

S

is a diffeomorphism.

Let : (, B; p; q1, . . . , qh) (, B; p; q1, . . . , qh)

be a strong deformation. Let (S, ), (S, ) be the symmetric Riemann surfaces

obtained by removing marked points from C, C, respectively. Define C : C C by

C(z) =

(z) if z (z) if z

Let denote the restriction of C to S. Then : (S, ) (S, ) is a strongdeformation.

Given , > 0 and

= [(, B; p; q1, . . . , qh)]

M(g,h),(n,m),

we will define a neighborhood U(,,) of in M(g,h),(n,m). Let (S, ) be thesymmetric Riemann surface obtained by removing marked points from C. Let

= [(, B; p; q1, . . . , qh)] M(g,h),(n,m),and (S, ) be the associated symmetric Riemann surface. Then M(g,h),(n,m) if

(1) There exists a -invariant oriented geodesic decomposition of S into pairsof pants.

(2) There exists a strong deformation

: (, B; p; q1, . . . , qh) (, B; p; q1, . . . , qh)i n the sense of Definition 4.8. So we have a strong deformation : S Sas above.

(3) Let lj , j and lj , j be the Fenchel-Nielsen coordinates for P and (P),respectively. Set d = 6g + 3h 6 + 2n + m. We have |lj lj| < for

j = 1, . . . , d, and |j j | < if lj > 0.

{U(,,) | , > 0, M(g,h),(n,m)}form a basis of the Fenchel-Nielsen topology.

U(,,) can be described more precisely. Set zj = ljeij , then up to permutationand complex conjugation of some zk we have

(z1, z2, . . . , z2d11, z2d1 , z2d1+1, . . . , zd) = ( z2, z1, . . . , z2d, z2d1, z2d1+1, . . . , zd),

so the fixed locus of consists of points of the form

(z2, z2, . . . , z2d11, z2d1, x1, . . . , xd2),where 2d1 + d2 = d, z2, z4, . . . , z2d1 C, and x1, . . . , xd2 R. The coordinatestake values in the fixed locus of , and xi are nonnegative on M(g,h),(n,m) becausenegative values correspond to nonorientable surfaces, as we have seen in Section 3.3.We conclude that U(,,) is homeomorphic to U/, where U is an open subsetofCd1 [0, )d2, and is the automorphism group of . The transition functionsbetween charts are real analytic ([Wolp, Appendix]), so Fenchel-Nielsen coordinatesgive M(g,h),(n,m) the structure of an orbifold with corners. The topology determined


31/85

31

by the structure of an orbifold with corners coincides with Fenchel-Nielsen topology.Therefore, we may equip M(g,h),(n,m) with a metric which induces Fenchel-Nielsontopology. In particular, the topology is Hausdorff, and compactness is equivalentto sequential compactness. A straightforward generalization of the argument in[Se, Section 6] shows that M(g,h),(n,m) is sequentially compact in Fenchel-Nielsentopology. Therefore,

Theorem 4.9. M(g,h),(n,m) is Hausdorff and compact in Fenchel-Nielson topology.

4.5. Orientation. M(g,h),(n,m) is an orbifold with corners, so we may ask if it isorientable as an orbifold. By Stasheffs results in [St], we have

Theorem 4.10. M(0,1),(0,(m)) has (m1)! isomorphic connected components, whichcorrespond to the cyclic ordering of the m boundary marked points. Each connectedcomponent of M(0,1),(0,(m)) is homeomorphic to R

m3.

Lemma 4.11. Suppose that (g, h, n)

= (0, 1, 0), and mi > 0. If M(g,h),(n,m) is

orientable, then M(g,h),(n,(m1,...,mi+1,...,mh)) is orientable.

Proof. Assume that M(g,h),(n,(m1,...,mi,...,mh))) is orientable. Consider the map

(1) F : M(g,h),(n,(m1,...,mi+1,...,mh)) M(g,h),(n,m),given by forgetting the last boundary marked point on the i-th boundary circle.Under our assumption, the fiber of F over [(, B; p; q1, . . . , qh)] is a union of mi

intervals and inherits the orientation of Bi. So M(g,h),(n,(m1,...,mi+1,...,mh))) is ori-entable. 2

Lemma 4.12. Supposet that (g, h, n) = (0, 1, 0), and mi = 0. IfM(g,h),(n,(m1,...,mi+1,...,mh))) is orientable, then M(g,h),(n,m) is orientable.

Proof. Assume that M(g,h),(n,(m1,...,mi+1,...,mh))) is orientable. Let T be the tangent

bundle of M(g,h),(n,m), which is an orbibundle over M(g,h),(n,m). To show thatM(g,h),(n,m) is orientable, it suffices to show that the restriction of T to every loop

in M(g,h),(n,m) is orientable. Let N(g,h),(n,m) be the interior of M(g,h),(n,m). Moreprecisely, N(g,h),(n,m) corresponds to surfaces with no boundary nodes. Since every

loop in M(g,h),(n,m) is homotopic to a loop in N(g,h),(n,m), it suffices to show thatN(g,h),(n,m) is orientable.

Suppose that = [(, B; p; q1, . . . , qh)] N(g,h),(n,m). Then Bi is an embeddedcircle in , oriented as in Remark 2.13, and the fiber of the map F in (1) over can be identified with Bi. So N(g,h),(n,m) is orientable. 2

It is shown in [IS2] that

Theorem 4.13. Suppose that (g, h, n)

= (0, 1, 0). Then M(g,h),(n,(1,...,1)) is a com-

plex orbifold.

Theorem 4.10, Lemma 4.11, Lemma 4.12, and Theorem 4.13 imply that

Theorem 4.14. M(g,h),(n,m) is orientable.

Let Q(g,h),n be the moduli space of stable bordered Riemann surfaces of type

(g, h) with n interior points. There is an h! to one map M(g,h),(n,0) Q(g,h),n, givenby forgetting the ordering of boundary components. Then Q(g,h),n is nonorientable.


32/85

32

1

2

3 4

Figure 18. Q(g,h),n is nonorientable

For example, consider M(1,2),(0,0) represented by the surface as show inFigure 18. The local coordinates are (l1, 1, l2, 2, l3, l4), where lj is the length ofj for j = 1, . . . , 4, and 1, 2 are gluing angles for 1, 2, respectively. There is

an automorphism of order 2 of which rotates the above Figure 14 by 180.(1) = 2, (3) = 4, and (l1, 1, l2, 2, l3, l4) = (l2, 2, l1, 1, l4, l3), whichis orientation reversing.

In general, an automorphism induces permutation ofd1 decomposing curves andpermutation of d2 bordered curves. The former corresponds to permutation ofpairs (lj , j), j = 1, . . . , d1, which is orientation preserving. The later correspondsto permutation of (ld1+1, . . . , ld1+d2) which is orientation preserving if and only if itis an even permutation. When we consider M(g,h),(n,0), automorphisms permuting

border curves are not allowed.

5. Moduli Space of Stable Maps

5.1. Prestable and stable maps. Let (X, ) be a compact symplectic manifold,

and let L be a Lagrangian submanifold. Let J be an -tame almost complexstructure.

Definition 5.1. A prestable map is a continuous map u : (, ) (X, L) suchthat J du = du j, where is a prestable bordered Riemann surface, u = u , : is the normalization map (Definition 3.7).Definition 5.2. A prestable map of type (g, h) with (n, m) marked points consistsof a prestable marked bordered Riemann surface of type (g, h) with (n, m) markedpoints (, B; p; q1, . . . , qh) and a prestable map u : (, ) (X, L).Definition 5.3. A morphism between prestable maps of type (g, h) with (n, m)marked points

(, B; p; q1, . . . , qh; u)

(, B; p; (q)1, . . . , (q)h; u)

is an isomorphism

: (, B; p; q1, . . . , qh) (, B; p; (q)1, . . . , (q)h)between prestable marked bordered Riemann surfaces of type (g, h) with(n, m) pointssuch that u = u .Definition 5.4. A prestable map of type (g, h) with(n, m) marked points is stableif its automorphism group is finite.


33/85

33

5.2. C topology. Let H2(X, L;Z), = (1, . . . , h) H1(L;Z)h be suchthat 1 + + h = , where : H2(X, L;Z) H1(L;Z) is the connectingmap in the long exact sequence for relative homology groups. Let h be a positiveinteger, g, n be nonnegative integers, m = (m1, . . . , mh) be an h-uple of nonnegativeintegers, and be an integer. Given the above data, define

M(g,h),(n,m)(X, L | , , )to be the moduli space of isomorphism classes of stable maps of type ( g, h) with(n, m) marked points

(, B; p; q1, . . . , qh; u)

such that u[] = , u[Bi] = i for i = 1, . . . , h, and (uT X , uT L) = . Here(uT X , uT L) is the Maslov index defined in [KL, Definition 3.3.7, Definition3.7.2]. From now on, we assume that L is oriented, so (uT X , uT L) is even, andwe may restrict ourselves to even . We will also assume that none of the i iszero, so the domain cannot have boundary nodes of type E.

Let M(g,h),(n,m)(X, L) be the moduli space of isomorphism classes of stable mapsof type (g, h) with (n, m) marked points. Then M(g,h),(n,m)(X, L | , , ) aredisjoint subsets of M(g,h),(n,m)(X, L) for different (, , ).

Let be a prestable bordered Riemann surface, and let : be thenormalization. Let u : (, ) (X, L) be a continuous map such that u = u :(, ) (X, L) is C w.r.t. g0 on X and some Hermitian metric h on , l 1.Define

a(u) =

ux2 uy

2 ux , uy2

g(x, y)dx dy,

where (x, y) are local isothermal coordinates on , and g(x, y)dxdy is the volumeform for the metric h. If u is an embedding, a(u) is the area of u() w.r.t. g0. If u

is a prestable map, then a(u) = 12 du L2= (u[]) [], where

du 2L2=

ux2 + uy

2

g(x, y)dx dy.

(u[]) [] only depends on the relative homology class u[] H2(X, L;Z), sowe have a function a : M(g,h),(n,m) [0, ), which takes the constant value []on M(g,h),(n,m)(X, L | , , ).

With the above definition, M(g,h),(n,m)(X, L | , , ) is a set. We will equip itwith the structure of a topological space, and show that it is sequentially compactand Hausdorff in this topology. This topology was introduced by Gromov [Gr].

We want to say two stable maps are close if the complex structures on the domainare close, and the maps are close. To measure the closeness, we use metrics on the

domain and on the target. For the target X, J is an -tame complex structure,so g0(X, Y) =

12 ((X, JY) + (Y, JX)) is a Riemannian metric on X such that J

is an isometry. For the domain, by a Hermitian metric h on a prestable borderedRiemann surface we mean a Hermitian metric on h on , the normalization of.

We now introduce some notation. Let : be the normalization map.Given a node r and a small positive number , let B(r) = (B(r1) B(r2)),where 1(r) = {r1, r2}, and B(r) is the geodesic ball of radius for = 1, 2. Let


34/85

34

be sufficiently small so that B(r) are disjoint for r sing, where sing denotesthe set of nodes on . Set N() =

rsing B(r), K() = N().

Definition 5.5 (C topology). Let = (, B; p; q1, . . . , qh; u) be a prestable mapof type (g, h) with (n, m) marked points. For a Hermitian metric h on and1, . . . , 4 > 0, a neighborhood U(,h,1, . . . , 4) of u in M(g,h),(n,m)(X, L) in theC topology is defined as follows. A prestable map

= (, B; p; (q)1, . . . , (q)h; u)

belongs to U(,h,1, . . . , 4) if

(1) There is a strong deformation

: (, B; p; (q)1, . . . , (q)h) (, B; p; q1, . . . , qh)such that 1 is defined on K1().

(2) j (1)j C(K1())< 2, where j, j are complex structures on ,, respectively.

(3) u u 1 C(K1 ())< 3.(4) |a(u) a(u)| < 4.

(1) says that can be obtained by deforming , or equivalently, is in thesame or a higher stratum in M(g,h),(n,m), the moduli space of prestable markedbordered Riemann surfaces of type (g, h) with (n, m) marked points. (2) says that

(, B; p; q1, . . . , qh), (, B; p; (q)1, . . . , (q)h)

are close in the C topology (Definition 5.6). (3) says that the maps u, u are C

close away from the nodes. (4) implies that a : M(g,h),(n,m)(X, L) [0, ) is acontinuous function.

(, B; p; q1, . . . , qh; u), (, B; p; (q)1, . . . , (q)h; u)

represent the same point in M(g,h),(n,m)(X, L) if and only if there is a Hermitianmetric h on such that

(1) There is a homeomorphism

: (, B; p; (q)1, . . . , (q)h) (, B; p; q1, . . . , qh)which induces a diffeomorphism .

(2) j (1)j C()= 0, where j, j are complex structures on , ,respectively.

(3) u u 1 C()= 0.(4) |a(u) a(u)| = 0.

So the C topology is actually a topology on the moduli space M(g,h),(n,m)(X, L).M(g,h),(n,m)(X, L | , , ) is a closed subspace ofM(g,h),(n,m)(X, L) and is equippedwith the subspace topology.

When X is a point, we get the C topology of M(g,h),(n,m).

Definition 5.6. Let = (, B; p; q1, . . . , qh) be a stable (prestable) bordered Rie-mann surface of type (g, h) with (n, m) marked points. For a Hermitian metrich on and 1, 2 > 0, a neighborhood U(,h,1, 2) of u in M(g,h),(n,m) in theC topology is defined as follows. A stable (prestable) bordered Riemann surface = (, B; p; (q)1, . . . , (q)h) belongs to U(,h,1, 2) if


35/85

35

(1) There is a strong deformation : such that 1 is defined onK1().

(2)

j

(

1)j C(K1())

< 2

, where j, j

are complex structures on ,

, respectively.

5.3. Compactness and Hausdorffness. The following is the main theorem ofthis section.

Theorem 5.7. M(g,h),(n,m)(X, L | , , ) is Hausdorff and sequentially compactin the C topology.

The Hausdorffness can be proven as in the case of curves without boundary, seee.g. [Sie1, Proposition 3.8]. The compactness is a consequence of the followingtheorem.

Theorem 5.8 (Gromovs Compactness Theorem). Let{l} be a sequence inM(g,h),(nm)(X, L) such that a(l) < C for all l N. Then there is a subsequenceof {l} convergent in the C topology.

Gromovs compactness theorem [Gr, 1.5] for J-holomorphic curves without bound-ary was carried out in details in [PW, Ye]. The case with boundary was proved in[Ye] (see also [IS1, IS2]). In [Ye], the moduli space is compactified by the modulispace of cusp curves, or prestable maps in this paper. We will describe how theproof in [Ye] gives Theorem 5.8.

The C topology can be equivalently defined as follows.

Definition 5.9 (C Topology). A sequence

l = (l, Bl; pl; q1l , . . . , q

hl ; ul)

converges to = (, B; p; q1, . . . , qh; u) in the C topology if for each 1, . . . , 4 >0, there is an integer N such that for l N,

(1) There is a strong deformation l : l such that 1

l is defined onK1().(2) j (1l )jl C(K1())< 2.(3) u ul 1l C(K1 ())< 3.(4) |a(u) a(ul)| < 4.

Recall that M(g,h),(n,m) denotes the moduli space of prestable bordered Rie-mann surfaces of type (g, h) with (n, m) marked points. There is a map F :

M(g,h),(n,m)(X, L | , , ) M(g,h),(n,m), given by forgetting the map. M(g,h),(n,m)has infinitely many strata since one can keep on going to lower and lower strata byadding non-stable components spheres and discs.

We claim that the image of F is covered by only finitely many strata, or equiv-alently:

Lemma 5.10. The domains in M(g,h),(n,m)(X, L | , , ) have only finitely manytopological types.

Proof. There is a map M(g,h),(n,m) M(g,h),(n,m), given by contracting non-stablecomponents. Since a stable bordered Riemann surfaces of type (g, h) with (n, m)points can have only finitely many possible topological types, it suffices to get anupper bound for the number of non-stable irreducible components. The restrictionof a stable map to a non-stable irreducible component is nonconstant, so there is


36/85

36

a lower bound > 0 for the area of the restriction of the map to each non-stablecomponent by [Ye, Lemma 4.3, Lemma 4.5]. Therefore, the number of non-stableirreducible components cannot exceed (

[])/. 2

Let {l} be a sequence in M(g,h),(n,m)(X, L) such that a() < C for all l N. ByLemma 5.10, there is a subsequence of{l} such that the domains are of the sametopological type. By normalization we obtain several sequences of stable maps withsmooth domains of the same topological type and with uniform area bound. Notethat each node gives rise to two marked points on the normalization. It suffices toshow that each sequence has a subsequence convergent in the C topology. So wemay assume that the domain is a smooth marked bordered Riemann surface or asmooth curve with marked points. In this case, it is proven in [Ye] that there is asubsequence which converges to a prestable map in the C topology. However, itis straightforward to check that the limit produced in [Ye] is actually a stable map.

6.Construction of Kuranishi Structure

6.1. Kuranishi structure with corners. We first quote the following definitionfrom [FO3, A2.1.1-A2.1.4], which is a slight modification of [FO, Definition 5.1].

Definition 6.1 (Kuranishi neighborhood). LetM be a Hausdorff topological space.A Kuranishi neighborhood (with corners) of p M is a 5-uple (Vp, Ep, p, p, sp)such that

(1) Vp is a smooth manifold (with corners), and Ep is a smooth vector bundleon it.

(2) p is a finite group which acts smoothly on Ep Vp.(3) sp is a p-equivariant continuous section of Ep.(4) p : s1p (0) M is a continuous map which induces a homeomorphism

from s1p (0)/p to a neighborhood of p in M.We call Ep the obstruction bundle and sp the Kuranishi map.

The following equivalence relation is weaker than the one in [FO, Definition 5.2],so the resulting equivalence class is larger.

Definition 6.2. Let M be a Hausdorff topological space. Two Kuranishi neigh-borhoods (with corners) (V1,p, E1,p, 1,p, 1,p, s1,p) and(V2,p, E2,p, 2,p, 2,p, s2,p) of

p M are equivalent if(1) dim V1,p rankE1,p = dim V2,p rankE2,p d.(2) There is another Kuranishi neighborhood (with corners) (Vp, Ep, p, p, sp)

of p such that dim Vp rankEp = d.(3) There are homomorphisms hi : i,p p for i = 1, 2.(4) For i = 1, 2, there is a i,p-invariant open neighborhood Vi of

1i,p (p), an

hi-equivariant embedding i : Vi Vp, and an hi-equivariant embedding ofvector bundles i : Ei,p|Vi Ep which covers i.

(5) i si,p = sp i for i = 1, 2.(6) i,p = p i for i = 1, 2.

In this case, we write (V1,p, E1,p, 1,p, 1,p, s1,p) (V2,p, E2,p, 2,p, 2,p, s2,p)The following definition is a combination of [FO3, A2.1.5-A2.1.11] and [FO,

Definition 5.3].


37/85

37

Definition 6.3 (Kuranishi structure). Let M be a Hausdorff topological space.A Kuranishi structure (with corners) on M assigns a Kuranishi neighborhood (ora Kuranishi neighborhood with corners) (V

p, Ep

, p

, p

, sp

) to each p

M and a

4-uple (Vpq, pq, pq, hpq) to each pair (p,q) where p M, q p(s1p (0)) such that(1) Vp

Date post:	06-Apr-2018
Category:	Documents
Upload:	swertyy
View:	223 times
Download:	0 times

Chiu-Chu Melissa Liu- Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open...

Documents