Difference between revisions of "Moduli spaces of pseudoholomorphic polygons"

Revision as of 11:40, 28 May 2017

To construct the moduli spaces from which the composition maps are defined we fix an auxiliary almost complex structure $J:TM\to TM$ which is compatible with the symplectic structure in the sense that $\omega (\cdot ,J\cdot )$ defines a metric on $M$ . (Unless otherwise specified, we will use this metric in all following constructions.)

Then given Lagrangians $L_{0},\ldots ,L_{d}\subset M$ and generators $x_{0}\in {\text{Crit}}(L_{0},L_{d}),$ $x_{1}\in {\text{Crit}}(L_{0},L_{1}),\ldots ,$ $x_{d}\in {\text{Crit}}(L_{{d-1}},L_{d})$ of their morphism spaces, we need to specify the Gromov-compactified moduli space $\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ . (Here and throughout, we will call a moduli space Gromov-compact if its subsets of bounded symplectic area are compact in the Gromov topology.) We will do this by combining two special cases which we discuss first.

Pseudoholomorphic polygons for pairwise transverse Lagrangians

If each consecutive pair of Lagrangians is transverse, $L_{0}\pitchfork L_{1},L_{1}\pitchfork L_{2},\ldots ,L_{{d-1}}\pitchfork L_{d},L_{d}\pitchfork L_{0}$ , then our construction is based on pseudoholomorphic polygons

$u:\Sigma \to M,\qquad u((\partial \Sigma )_{i})\subset L_{i},\qquad \overline \partial _{J}u=0,$

where $\Sigma =\Sigma _{{\underline {z}}}:=D\setminus \{z_{0},\ldots ,z_{d}\}$ is a disk with $d+1$ boundary punctures in counter-clockwise order $z_{0},\ldots ,z_{d}\subset \partial D$ , and $(\partial \Sigma )_{i}$ denotes the boundary component between $z_{i},z_{{i+1}}$ (resp. between $z_{{d}},z_{0}$ for i=d). More precisely, we construct the (uncompactified) moduli spaces of pseudoholomorphic polygons for any tuple $x_{i}\in L_{i}\cap L_{{i+1}}$ for $i=0,\ldots ,d$ as in [Seidel book]:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(\underline {z},u)\,{\big |}\,{\text{1. - 3.}}{\bigr \}}/\sim$

where

$\underline {z}=(z_{0},z_{1},\ldots ,z_{d})\subset \partial D$ is a tuple of pairwise disjoint marked points on the boundary of a disk, in counter-clockwise order.
is a smooth map satisfying
- the Cauchy-Riemann equation $\overline \partial _{J}u=0$ ,
- Lagrangian boundary conditions $u((\partial \Sigma )_{i})\subset L_{i}$ ,
- the finite energy condition $\textstyle \int _{{\Sigma }}u^{*}\omega <\infty$ ,
- the limit conditions $\lim _{{z\to z_{i}}}u(z)=x_{i}$ for $i=0,1,\ldots ,d$ .
The pseudoholomorphic polygon $(\underline {z},u)$ is stable in the sense that the map $u:\Sigma _{{\underline {z}}}\to M$ is nonconstant if the number of marked points is $d+1<3$ .

Here two pseudoholomorphic polygons are equivalent $(\underline {z},u)\sim (\underline {z}',u')$ if there is a biholomorphism $\psi :D\to D$ that preserves the marked points $\psi (z_{i})=z'_{i}$ , and relates the pseudoholomorphic polygons by reparametrization, $u=u'\circ \psi$ .

The case $d=0$ is not considered in this part of the moduli space setup since $L_{0},L_{d}=L_{0}$ are never transverse. However, it might appear in the construction of homotopy units?

[Expand]

The domains of the pseudoholomorphic polygons are strips for $d=1$ and represent elements in a Deligne-Mumford space for $d\geq 2$ as follows:

All isotropy groups of this uncompactified moduli space

{\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})

are trivial; that is any disk biholomorphism

\psi :D\to D

that fixes

d+1\geq 1

marked points

\psi (z_{i})=z_{i}

, and preserves a pseudoholomorphic map

u=u\circ \psi

must be the identity

\psi ={{\rm {id}}}_{D}

.

[Expand]Proof:

Next, to construct the Gromov-compactified moduli spaces $\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ we have to add various strata to the moduli space of pseudoholomorphic polygons without breaking or nodes ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ defined above.

[Expand]

This is done precisely in the general construction below, but roughly requires to include breaking and bubbling, in particular

We will see that sphere bubbling does not contribute to the boundary stratification of these moduli spaces, so that the boundary stratification and thus the algebraic structure arising from these moduli spaces is induced by Floer breaking and disk bubbling. (On the other hand, sphere bubbling will be the only source of nontrivial isotropy.) The boundary strata arising from Floer breaking are fiber products of other moduli spaces of pseudoholomorphic polygons over finite sets of Lagrangian intersection points, which indicates an algebraic composition in this finitely generated Floer chain complex.

[Expand]

Disk bubbling, on the other hand, in the present setting yields boundary strata that are fiber products over the Lagrangian submanifold specified by the boundary condition, which is problematic for a combination of algebra and regularity reasons.

We will resolve this issue as in [J.Li thesis] by following another earlier proposal by Fukaya-Oh to allow disks to flow apart along a Morse trajectory, thus yielding disk trees which are constructed next - still ignoring sphere bubbling.

Pseudoholomorphic disk trees for a fixed Lagrangian

If the Lagrangians are all the same, $L_{0}=L_{1}=\ldots =L_{d}=:L$ , then our construction is based on pseudoholomorphic disks

$u:D\to M,\qquad u(\partial D)\subset L,\qquad \overline \partial _{J}u=0.$

Such disks (modulo reparametrization by biholomorphisms of the disk) also arise from Gromov-compactifying other moduli spaces of pseudoholomorphic curves in which energy concentrates at a boundary point. To capture this bubbling algebraically, we work throughout with the Morse function $f:L\to \mathbb{R}$ chosen in the setup of the morphism space ${\text{Hom}}(L,L)=\textstyle \sum _{{x\in {\text{Crit}}(f)}}\Lambda x$ . We also choose a metric on $L$ so that the gradient vector field $\nabla f$ satisfies the Morse-Smale conditions and an additional technical assumption in [1] which guarantees a natural smooth manifold-with-boundary-and-corners structure on the compactified Morse trajectory spaces $\overline {\mathcal {M}}(L,L),\overline {\mathcal {M}}(p^{-},L),\overline {\mathcal {M}}(L,p^{+}),\overline {\mathcal {M}}(p^{-},p^{+})$ for $p^{\pm }\in {\text{Crit}}(f)$ . This smooth structure is essentially induced by the requirement that the evaluation maps at positive and negative ends ${{\rm {ev^{\pm }}}}:\overline {\mathcal {M}}(\ldots )\to L$ are smooth. With that data and the fixed almost complex structure $J$ we can construct the moduli spaces of pseudoholomorphic disk trees for any tuple $x_{0},x_{1},\ldots ,x_{d}\in {{\rm {Crit}}}(f)$ as in JL:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {u})\,{\big |}\,{\text{1. - 5.}}{\bigr \}}/\sim$

where

1. $T$ is an ordered tree with sets of vertices $V=V^{m}\cup V^{c}$ and edges $E$ ,

[Expand]

equipped with orientations towards the root, orderings of incoming edges, and a partition into main and critical (leaf and root) vertices as follows:

2. $\underline {\gamma }=(\underline {\gamma }_{e})_{{e\in E}}$ is a tuple of generalized Morse trajectories for each edge

[Expand]

in the following compactified Morse trajectory spaces:

3. $\underline {z}=(\underline {z}_{v})_{{v\in V^{m}}}$ is a tuple of boundary marked points for each main vertex

[Expand]

that are ordered counter-clockwise as follows:

4. $\underline {u}=(\underline {u}_{v})_{{v\in V^{m}}}$ is a tuple of pseudoholomorphic disks for each main vertex,

[Expand]

that is each $v\in V^{m}$ is labeled by a smooth map $u_{v}:D\to M$ satisfying Cauchy-Riemann equation, Lagrangian boundary condition, finite energy, and matching conditions as follows:

5. The disk tree is stable

[Expand]

in the sense that

Finally, two pseudoholomorphic disk trees are equivalent $(T,\underline {\gamma },\underline {z},\underline {u})\sim (T',\underline {\gamma }',\underline {z}',\underline {u}')$ if

[Expand]

there is a tree isomorphism $\zeta :T\to T'$ and a tuple of disk biholomorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ which preserving the tree, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that

[Expand]

The domains of the disk trees are never stable for $d=0,1$ , but need to be studied to construct the differential $\mu ^{1}$ on the Floer chain complex and the curvature term $\mu ^{0}$ that may obstruct $\mu ^{1}\circ \mu ^{1}=0$ . For $d\geq 2$ the domains of the disk trees represent elements in a Deligne-Mumford space as follows:

Any equivalence class of disk trees $[(T,\underline {\gamma },\underline {z},\underline {u})]$ induces a domain tree $[(T',\underline {\ell },\underline {z})]$

[Expand]

by forgetting the Morse trajectories and pseudoholomorphic maps as follows:

For $d\geq 2$ , such a domain tree is called stable if every vertex has valence $|v|\geq 3$ - i.e. there are at least three marked points on each disk $D_{v}$ . The domain trees for $d=0,1$ are never stable, but both cases need to be included in our moduli space constructions: The differential $\mu ^{1}$ on the Floer chain complex $Hom(L,L)=\textstyle \sum _{{x\in {{\rm {Crit}}}(f)}}\Lambda \;x$ is constructed by counting the elements of ${\mathcal {M}}(x_{0};x_{1})$ . The curvature term $\mu ^{0}$ , which is constructed from moduli spaces ${\mathcal {M}}(x_{0})$ with no incoming critical points, serves to algebraically encode disk bubbling in any moduli space involving a Lagrangian boundary condition on $L$ .

For $d\geq 2$ , while the above trees are not necessarily stable, they induce unique stable rooted metric ribbon trees $(T,\underline {\ell })$ in the sense of [Def.2.7, MW], by forgetting the marked points, forgetting every leaf of valence 1 and its outgoing edge, and replacing every vertex $v$ of valence 2 and its incoming and outgoing edges $(v^{-},v),(v,v^{+})$ by a single edge $(v^{-},v^{+})$ of length $\ell _{{(v^{-},v^{+})}}=\ell _{{(v^{-},v)}}+\ell _{{(v,v^{+})}}$ . The space of such stable rooted metric ribbon trees - where a tree containing an edge of length $\ell _{e}=0$ is identified with the tree in which this edge and its adjacent vertices are replaced by a single vertex - is another topological representation of the Deligne Mumford space $\overline {\mathcal {M}}_{{d+1}}$ , as discussed in [BV]. Its boundary strata are given by trees with interior edges of length $\ell _{e}=\infty$ .

We now expect the boundary stratification of the moduli spaces of disk trees ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ - if/once regular - to arise exclusively from breaking of the Morse trajectories representing edges of the disk trees. This is made rigorous in [J.Li thesis] under the assumption that the almost complex structure $J$ can be chosen such that there exist no nonconstant $J$ -holomorphic spheres in the symplectic manifold $M$ . In that special case, all isotropy groups are trivial by [Prop.2.5, J.Li thesis]; that is any equivalence between a disk tree and itself, $(T,\underline {\gamma },\underline {z},\underline {u})\sim (T,\underline {\gamma },\underline {z},\underline {u})$ , is given by the trivial tree isomorphism $\zeta :T\to T$ , and the only disk biholomorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ which preserve the marked points and pseudoholomorphic disk maps are the identity maps $\psi _{v}={{\rm {id}}}_{D}$ . In this case, the moduli spaces of disk trees ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ are moreover Gromov-compact since sphere bubbling is ruled out and disk bubbling is captured by edges labeled with constant, zero length, Morse trajectories.

In general, we will Gromov-compactify ${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ in the following general construction by allowing for sphere bubble trees developing at any (boundary or interior) point of each of the disk domains. This will also be a source of generally nontrivial isotropy.

General moduli space of pseudoholomorphic polygons

For the construction of a general $A_{\infty }$ -composition map we are given $d+1\geq 1$ Lagrangians $L_{0},\ldots ,L_{d}\subset M$ and a fixed autonomous Hamiltonian function $H_{{L_{i},L_{j}}}:M\to \mathbb{R}$ for each pair $L_{i}\neq L_{j}$ whose time-1 flow provides transverse intersections $\phi _{{L_{i},L_{j}}}(L_{i})\pitchfork L_{j}$ . To simplify notation for consecutive Lagrangians in the list, we index it cyclically by $i\in \mathbb{Z } _{{d+1}}$ and abbreviate $\phi _{i}:=\phi _{{L_{{i-1}},L_{i}}}$ so that we have $\phi _{i}(L_{{i-1}})\pitchfork L_{i}$ whenever $L_{{i-1}}\neq L_{i}$ , and in particular $\phi _{0}(L_{d})\pitchfork L_{0}$ unless $L_{d}=L_{0}$ . Now, given generators $x_{0}\in {\text{Crit}}(L_{0},L_{d}),x_{1}\in {\text{Crit}}(L_{0},L_{1}),\ldots ,x_{d}\in {\text{Crit}}(L_{{d-1}},L_{d})$ of these morphism spaces, we construct the Gromov-compactified moduli space of pseudoholomorphic polygons by combining the two special cases above:

${\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}):={\bigl \{}(T,\underline {\gamma },\underline {z},\underline {u})\,{\big |}\,{\text{1. - 6.}}{\bigr \}}/\sim$

where

1. $T$ is an ordered tree with sets of vertices $V=V^{m}\cup V^{c}$ and edges $E$ ,

[Expand]

equipped with orientations towards the root, orderings of incoming edges, and a partition into main and critical (leaf and root) vertices as follows:

2. The tree structure induces tuples of Lagrangians $\underline {L}=(\underline {L}^{v})_{{v\in V^{m}}}$

[Expand]

that label the boundary components of domains in overall counter-clockwise order $L_{0},\ldots ,L_{d}$ as follows:

3. $\underline {\gamma }=(\underline {\gamma }_{e})_{{e\in E}}$ is a tuple of generalized Morse trajectories

[Expand]

in the following compactified Morse trajectory spaces:

4. $\underline {z}=(\underline {z}_{v})_{{v\in V^{m}}}$ is a tuple of boundary points

[Expand]

that are ordered counter-clockwise and associate complex domains $\Sigma ^{v}:=D\setminus \underline {z}_{v}$ to the vertices as follows:

5. $\underline {u}=(\underline {u}_{v})_{{v\in V^{m}}}$ is a tuple of pseudoholomorphic maps for each main vertex,

[Expand]

that is each $v\in V^{m}$ is labeled by a smooth map $u_{v}:\Sigma ^{v}\to M$ satisfying Cauchy-Riemann equation, Lagrangian boundary conditions, finite energy, and matching conditions as follows:

6. The disk tree is stable

[Expand]

in the sense that

Finally, two pseudoholomorphic disk trees are equivalent $(T,\underline {\gamma },\underline {z},\underline {u})\sim (T',\underline {\gamma }',\underline {z}',\underline {u}')$ if

[Expand]

there is a tree isomorphism $\zeta :T\to T'$ and a tuple of disk biholomorphisms $(\psi _{v}:D\to D)_{{v\in V^{m}}}$ which preserve the tree, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that

[Expand]

NOTE: when degenerating polygons to create a strip with $L_{i}=L_{j}$ boundary conditions, will need to transfer from Morse-Bott breaking to boundary node

Finally, the symplectic area function $\omega :\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})\to \mathbb{R}$ in each case is given by TODO

Fredholm index ${\mathcal {M}}^{k}(\ldots )=\{b\in {\mathcal {M}}(\ldots )\,|\,IND...=k\}$

@@ Line 31: / Line 31: @@
 # The pseudoholomorphic polygon <math>(\underline{z}, u )</math> is stable in the sense that the map <math>u:\Sigma_{\underline{z}}\to M</math> is nonconstant if the number of marked points is <math>d+1<3</math>.
-Here two pseudoholomorphic polygons are equivalent <math>(\underline{z}, u) \sim (\underline{z}', u')</math> if there is a disk automorphism <math>\psi:D\to D</math> that preserves the complex structure on <math>D</math>, the marked points <math>\psi(z_i)= z'_i</math>, and relates the pseudoholomorphic polygons by reparametrization, <math>u = u'\circ\psi</math>.
+Here two pseudoholomorphic polygons are equivalent <math>(\underline{z}, u) \sim (\underline{z}', u')</math> if there is a biholomorphism <math>\psi:D\to D</math> that preserves the marked points <math>\psi(z_i)= z'_i</math>, and relates the pseudoholomorphic polygons by reparametrization, <math>u = u'\circ\psi</math>.
 ----
@@ Line 49: / Line 49: @@
 </div></div>
-All isotropy groups of this uncompactified moduli space <math>\mathcal{M}(x_0;x_1,\ldots,x_d)</math> are trivial; that is any disk automorphism <math>\psi:D\to D</math> that fixes <math>d+1\geq 1</math> marked points <math>\psi(z_i)= z_i</math>, and preserves a pseudoholomorphic map <math>u = u\circ\psi</math> must be the identity <math>\psi={\rm id}_D</math>. <div class="toccolours mw-collapsible mw-collapsed">'''Proof:'''<div class="mw-collapsible-content"> In case <math>d=2</math> this follows directly from the marked points, since any Mobius transformation that fixes three points is the identity.
+All isotropy groups of this uncompactified moduli space <math>\mathcal{M}(x_0;x_1,\ldots,x_d)</math> are trivial; that is any disk biholomorphism <math>\psi:D\to D</math> that fixes <math>d+1\geq 1</math> marked points <math>\psi(z_i)= z_i</math>, and preserves a pseudoholomorphic map <math>u = u\circ\psi</math> must be the identity <math>\psi={\rm id}_D</math>. <div class="toccolours mw-collapsible mw-collapsed">'''Proof:'''<div class="mw-collapsible-content"> In case <math>d=2</math> this follows directly from the marked points, since any Mobius transformation that fixes three points is the identity.
-In case <math>d=1</math> this requires both the stability and finite energy conditions: The group of automorphisms that fix two marked points - i.e. the automorphisms of the strip - are shifts by <math>\R</math>.
+In case <math>d=1</math> this requires both the stability and finite energy conditions: The group of biholomorphisms that fix two marked points - i.e. the biholomorphisms of the strip - are shifts by <math>\R</math>.
 On the other hand, any J-holomorphic map <math>u:\R\times[0,1]\to M</math> has nonnegative energy density <math>u^*\omega= \bigl(|\partial_s u|^2+|\partial_t u|^2\bigr) ds\wedge dt</math> with <math>|\partial_s u|=|\partial_t u|</math>.
 If we now had nontrivial isotropy, i.e. <math>u(\tau+s,t)=u(s,t)</math> for some <math>\tau>0</math> and a nonconstant map <math>u</math>, then there would exist <math>s_0,t_0\in \R\times[0,1]</math> with <math>|\partial_s u(s_0,t_0)|=|\partial_t u(s_0,t_0)|>0</math> and thus
@@ Line 90: / Line 90: @@
 </math>
 </center>
-Such disks (modulo reparametrization by automorphisms of the disk) also arise from Gromov-compactifying other moduli spaces of pseudoholomorphic curves in which energy concentrates at a boundary point.
+Such disks (modulo reparametrization by biholomorphisms of the disk) also arise from Gromov-compactifying other moduli spaces of pseudoholomorphic curves in which energy concentrates at a boundary point.
 To capture this bubbling algebraically, we work throughout with the Morse function <math>f:L\to\R</math> chosen in the setup of the morphism space <math>\text{Hom}(L,L)=\textstyle\sum_{x\in\text{Crit}(f)} \Lambda x</math>. We also choose a metric on <math>L</math> so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions and an additional technical assumption in [https://arxiv.org/abs/1205.0713] which guarantees a natural smooth manifold-with-boundary-and-corners structure on the [[compactified Morse trajectory spaces]]
 <math> \overline\mathcal{M}(L,L) , \overline\mathcal{M}(p^-,L), \overline\mathcal{M}(L,p^+), \overline\mathcal{M}(p^-,p^+) </math> for <math>p^\pm\in\text{Crit}(f)</math>.
@@ Line 150: / Line 150: @@
 Finally, two pseudoholomorphic disk trees are equivalent <math>(T, \underline{\gamma}, \underline{z}, \underline{u} ) \sim (T', \underline{\gamma}', \underline{z}', \underline{u}')</math> if
 <div class="toccolours mw-collapsible mw-collapsed">
-there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk automorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math>
+there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk biholomorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math>
-which preserving the tree, complex structure, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that
+which preserving the tree, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that
 <div class="mw-collapsible-content">
 * <math>\zeta</math> preserves the tree structure and order of edges;
-* <math>\psi_v^* i = i</math>, where <math>i:TD\to TD</math> denotes the complex structure on <math>D</math>;
 *<math>\underline{\gamma}_e=\underline{\gamma}'_{\zeta(e)}</math> for every <math>e\in E</math>;
 * <math>\psi_v(z^v_e)= {z'}^{\zeta(v)}_{\zeta(e)}</math> for every <math>v\in V^m</math> and adjacent edge <math>e\in E_v</math>;
@@ Line 177: / Line 176: @@
 # Every edge <math>e\in E' = E</math> is labeled with the length <math>\ell_e:= \ell(\underline{\gamma}_e)\in[0,\infty]</math> of the associated generalized Morse trajectory. For the semi-infinite edges, this length is automatically <math>\ell_e=\infty</math> since the associated Morse trajectories are semi-infinite.
 # The domain for each vertex <math>v\in V'</math> is marked by <math>|v|</math> boundary points <math>\underline{z}_v=(z^v_e)_{e\in \{e^0_v\}\cup E^{\rm in}_v}\subset \partial D_v</math>, ordered counter-clockwise.
-# Two such trees are equivalent <math>[ (T, \underline{\ell} , \underline{z})]\sim [ (T', \underline{\ell}' , \underline{z}')]</math> if there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk automorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> preserving the complex structure on <math>D</math> such that <math>\zeta</math> preserves the ordered tree structure and lengths <math>\ell_e=\ell'_{\zeta(e)}</math> for every <math>e\in E</math>, and the marked points are preserved <math>\psi_v(z^v_e)= z{'\;\zeta(v)}_{\zeta(e)}</math> for every <math>v\in V^m</math> and adjacent <math>e\in E</math>.
+# Two such trees are equivalent <math>[ (T, \underline{\ell} , \underline{z})]\sim [ (T', \underline{\ell}' , \underline{z}')]</math> if there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk biholomorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> such that <math>\zeta</math> preserves the ordered tree structure and lengths <math>\ell_e=\ell'_{\zeta(e)}</math> for every <math>e\in E</math>, and the marked points are preserved <math>\psi_v(z^v_e)= z{'\;\zeta(v)}_{\zeta(e)}</math> for every <math>v\in V^m</math> and adjacent <math>e\in E</math>.
 </div>
 </div>
@@ Line 193: / Line 192: @@
 We now expect the boundary stratification of the moduli spaces of disk trees <math>\mathcal{M}(x_0;x_1, \ldots, x_d)</math> - if/once regular - to arise exclusively from breaking of the Morse trajectories representing edges of the disk trees. This is made rigorous in [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf J.Li thesis]] under the assumption that the almost complex structure <math>J</math> can be chosen such that there exist no nonconstant <math>J</math>-holomorphic spheres in the symplectic manifold <math>M</math>.
 In that special case, all isotropy groups are trivial by [[https://math.berkeley.edu/~katrin/papers/disktrees.pdf Prop.2.5, J.Li thesis]]; that is any equivalence between a disk tree and itself,
-<math>(T, \underline{\gamma}, \underline{z}, \underline{u}) \sim (T, \underline{\gamma}, \underline{z}, \underline{u})</math>, is given by the trivial tree isomorphism <math>\zeta:T\to  T</math>, and the only disk automorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> which preserve the marked points and pseudoholomorphic disk maps are the identity maps <math>\psi_v={\rm id}_D</math>.
+<math>(T, \underline{\gamma}, \underline{z}, \underline{u}) \sim (T, \underline{\gamma}, \underline{z}, \underline{u})</math>, is given by the trivial tree isomorphism <math>\zeta:T\to  T</math>, and the only disk biholomorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> which preserve the marked points and pseudoholomorphic disk maps are the identity maps <math>\psi_v={\rm id}_D</math>.
 In this case, the moduli spaces of disk trees <math>\mathcal{M}(x_0;x_1, \ldots, x_d)</math> are moreover Gromov-compact since sphere bubbling is ruled out and disk bubbling is captured by edges labeled with constant, zero length, Morse trajectories.
@@ Line 290: / Line 289: @@
 Finally, two pseudoholomorphic disk trees are equivalent <math>(T, \underline{\gamma}, \underline{z}, \underline{u} ) \sim (T', \underline{\gamma}', \underline{z}', \underline{u}')</math> if
 <div class="toccolours mw-collapsible mw-collapsed">
-there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk automorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> which preserve the tree, complex structure, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that
+there is a tree isomorphism <math>\zeta:T\to  T'</math> and a tuple of disk biholomorphisms <math>(\psi_v:D\to D)_{v\in V^m}</math> which preserve the tree, Morse trajectories, marked points, and pseudoholomorphic curves in the sense that
 <div class="mw-collapsible-content">
 * <math>\zeta</math> preserves the tree structure and order of edges;
-* <math>\psi_v^* i = i</math>, where <math>i:TD\to TD</math> denotes the complex structure on <math>D</math>;
 *<math>\underline{\gamma}_e=\underline{\gamma}'_{\zeta(e)}</math> for every <math>e\in E</math>;
 * <math>\psi_v(z^v_e)= {z'}^{\zeta(v)}_{\zeta(e)}</math> for every <math>v\in V^m</math> and adjacent edge <math>e\in E_v</math>;

Difference between revisions of "Moduli spaces of pseudoholomorphic polygons"

Revision as of 11:40, 28 May 2017

Pseudoholomorphic polygons for pairwise transverse Lagrangians

Pseudoholomorphic disk trees for a fixed Lagrangian

General moduli space of pseudoholomorphic polygons

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