Non-fillability of overtwisted contact manifolds via polyfolds

Following [50] , [56] we define:

The pages \(\vartheta ^{-1}(\theta )\) in \(N\setminus B\) are co-oriented by the orientation of \(S^1\), i.e. the linearisation \(T\vartheta \) maps positive normal vectors to positive tangent vectors of \(S^1\).

As in [50], Section 4 and [56], Section I.4 we define:

In particular, the binding of a bordered Legendrian open book is an isotropic submanifold of \((M,\xi )\); the boundary \(\partial N\) is Legendrian. The complement

\[ N^*:=N\setminus \big (B\cup \partial N\big ) \]

of \(B\cup \partial N\) in \(N\) is the set of regular points of \(N\subset (M,\xi )\), i.e. the set of all points \(p\in N\) such that \(T_pN\) and \(\xi _p\) intersect transversally. The characteristic distribution \(TN^*\cap \xi \) integrates by the Frobenius theorem to the so-called characteristic foliation on \(N^*\). The characteristic leaves , which by definition are the leaves of the characteristic foliation, coincide with the pages of the open book \((B,\vartheta )\).

If, in addition, \(\xi \) is co-oriented, then \(\xi |_{N^*}\) puts a co-orientation to the pages \(\vartheta ^{-1}(\theta )\) in \(N\setminus B\). We will assume that this co-orientation coincides with the co-orientation induced by \(\vartheta \) by possibly composing \(\vartheta \) with a reflection on \(S^1=\partial D\).

It follows from [56], Theorem I.1.3 or [44], Theorem 1.4 that the germ of a contact structure \((M,\xi )\) is unique near a submanifold \(N\subset (M,\xi )\) (with boundary \(\partial N\)) that carries a bordered Legendrian open book \((B,\vartheta )\). The germ is uniquely determined by the singular characteristic distribution \(\xi \cap TN\) given by the open book decomposition on \(N\) determined by \((B,\vartheta )\).

A bordered Legendrian open book \((B,\vartheta )\) is called small if the supporting submanifold \(N\subset (M,\xi )\) is contained in a ball inside \(M\).

A \((2n-1)\)-dimensional contact manifold \((M,\xi )\) is called overtwisted , if \((M,\xi )\) contains an overtwisted disc, see [11] . For example \(\R ^3\) equipped with the contact structure \(\ker \alpha _{\ot }\),

\[ \alpha _{\ot }:=\cos r\,\rmd z+r\sin r\,\rmd \theta \;, \]

is overtwisted, as \(D_{\ot }^2:=\{z=0,r\leq \pi \}\) is an overtwisted disc.

For a \((n-2)\)-dimensional closed smooth manifold \(Q\) we consider the contact manifold \(\R ^3\times T^*Q\) equipped with contact structure \(\xi _Q:=\ker (\alpha _{\ot }+\lambda _{T^*Q})\), where we identify \(Q\) with the zero section in \(T^*Q\) and denote the Liouville \(1\)-form of \(T^*Q\) by \(\lambda _{T^*Q}\). Following [11], Section 10 we define the model plastikstufe with core \(Q\) to be the subset \(P_Q:=D_{\ot }^2\times Q\) of \(\big (\R ^3\times T^*Q,\xi _Q\big )\).

We will say that \((P_Q,\xi _Q)\) admits a contact embedding into a \((2n-1)\)-dimensional contact manifold \((M,\xi )\) if a neighbourhood of \(P_Q\) in \(\big (\R ^3\times T^*Q,\xi _Q\big )\) does. In this case the image \(N\) of the model \(P_Q\) is called a plastikstufe with core \(Q\) and carries the structure of a bordered Legendrian open book with binding \(Q\) and pages corresponding to \(I_{\theta }\times Q\), where \(I_{\theta }\) is the straight line segment in \(D_{\ot }^2\subset \R ^2\) connecting \(0\) and \(\pi \rme ^{\rmi \theta }\). For the original definition of a plastikstufe and the relation to bordered Legendrian open books we refer to [55] , [56] .

The assumption on \(Q\) is satisfied for any stably parallelisable manifold \(Q\), cf. [9], Section 1.1 . For the converse of Theorem 70 we note:

A forerunner version of this result, which is [45], Theorem 1.2 , was given in [17] . Further, it is shown in [45], Theorem 1.3 that if \((M,\xi )\) admits a bordered Legendrian open book and \(\dim M=5\), then \((M,\xi )\) is overtwisted. In fact, a contact manifold \((M,\xi )\) is overtwisted precisely if \((M,\xi )\) contains a bordered Legendrian open book with pages diffeomorphic to \(P\times \Sigma \), where \(P\) is a closed manifold and \(\Sigma \) a compact surface with boundary, see [45], Corollary 1.4 .

Let \((M,\xi )\) be a contact manifold. We consider a bordered Legendrian open book decomposition \((B,\vartheta )\) of a submanifold \(N\subset (M,\xi )\). By Definition 2.1.1 the binding \(B\subset N\) admits a tubular neighbourhood \(B\times D^2\) on which the fibre projection \(\vartheta \) is given by \((b,z=r\rme ^{\rmi \theta })\mapsto \theta \).

By [55], Proposition 4 a neighbourhood of \(B\times D^2\) in \((M,\xi )\) is contactomorphic to a neighbourhood of \(\{0\}\times D^2\times B\) in \(\big (\R \times \C \times T^*B,\ker \alpha _o\big )\), where

\[ \alpha _o:= \rmd t +\tfrac 12\big (x\rmd y-y\rmd x\big ) +\lambda _{T^*B}\;, \]

denoting by \(t,z=x+\rmi y\) the coordinates on \(\R \times \C \) and by \(\lambda _{T^*B}\) the Liouville \(1\)-form on \(T^*B\). The contactomorphism restricts to \((b,z)\mapsto (0,z,b)\) on \(B\times D^2\). Again we identify \(B\) with the zero section in \(T^*B\).

Consider a contact manifold \((M,\xi )\). Let \(N\subset (M,\xi )\) be a submanifold that supports a bordered Legendrian open book \((B,\vartheta )\). By Definition 2.1.1 the restriction of \(\vartheta \) to \(\partial N\) induces a locally trivial fibration over \(S^1\) with fibre \(F\). Denoting the monodromy diffeomorphism by \(\varphi \co F\ra F\) this fibration is equivalent to the mapping torus

\[ M(\varphi )= \frac {[0,2\pi ]\times F} {(2\pi ,f)\sim \big (0,\varphi (f)\big )} \]

of \(\varphi \). The induced diffeomorphism

\[ \varphi ^*:=(T\varphi ^{-1})^*\co T^*F\ra T^*F \]

naturally preserves the Liouville \(1\)-form \(\lambda _{T^*F}\) so that the mapping torus

\[ M(\varphi ^*)= \frac {T^*[0,2\pi ]\times T^*F} {\big ((r,2\pi ),u\big )\sim \big ((r,0),(T\varphi ^{-1})^*(u)\big )} \]

carries the Liouville form \(r\rmd \theta +\lambda _{T^*F}\) and can be identified with \(T^*\big (M(\varphi )\big )\). The corresponding Liouville vector field is of the form \(r\partial _r+Y_{T^*F}\) and \(M(\varphi ^*)\) fibres naturally over \((T^*S^1,r\rmd \theta )\) with fibre projection map \([(r,\theta ),u]\mapsto [(r,\theta )]\).

We equip \(\R \times M(\varphi ^*)\) with the contact form \(\alpha _{\varphi }\) induced by

\[ \alpha _{\varphi }\equiv \rmd t+r\rmd \theta +\lambda _{T^*F} \;. \]

By [50], Lemma 4.6 a neighbourhood of \(\partial N\subset (M,\xi )\) is contactomorphic to a neighbourhood of

\[ \{0\}\times M(\varphi ) \subset \big (\R \times M(\varphi ^*),\ker \alpha _{\varphi }\big ) \;, \]

so that

\[ \big (\{0\}\times \{r=0,u=0\}\big ) \equiv \{0\}\times M(\varphi ) \]

corresponds to \(\partial N\) and a neighbourhood of \(\partial N\) in \(N\subset M\) corresponds to the quotient of the set \(\{0\}\times \{r\leq 0,u=0\}\) in \(\{0\}\times M(\varphi ^*)\). We orient \(T^*S^1\equiv \R \times S^1\) by \(\rmd r\wedge \rmd \theta \). This matches the co-orientation conventions for the singular distribution determined by \(\xi \) and the pages of \((B,\vartheta )\) in Section 2.1 .

Motivated by Section 2.3 we define a natural almost complex structure \(J\) on \(\R \times (\R \times \C \times T^*B)\) that allows a lifting of obvious holomorphic discs similar to cf. [30], Section 2 .

For that choose a Riemannian metric \(g_B\) on \(B\). Denote by \(J_{T^*B}\) the almost complex structure on \(T^*B\) that is induced by the Levi-Civita connection of \(g_B\), see [55], Appendix B or [48], Section 5 . Observe that \(J_{T^*B}\) is compatible with the symplectic form \(\rmd \lambda _{T^*B}\). Furthermore denoting by \(g_B^{\flat }\) the dual metric of \(g_B\) the kinetic energy function on \(T^*B\) is defined by \(k(u)=\frac 12g^{\flat }(u,u)\), \(u\in T^*B\), and satisfies \(\lambda _{T^*B}=-\rmd k\circ J_{T^*B}\). In other words, \(k\) is a strictly plurisubharmonic potential in the sense of [28], Section 3.1 .

On the Liouville manifold

\[ (V,\lambda _V):= \Big ( \C \times T^*B,\tfrac 12\big (x\rmd y-y\rmd x\big ) +\lambda _{T^*B} \Big ) \]

we consider the almost complex structure

\[ J_V=\rmi \oplus J_{T^*B} \;, \]

which is compatible with the symplectic form \(\rmd \lambda _V\). The function

\[ \psi (z,u):=\tfrac 14|z|^2+k(u) \]

is a strictly plurisubharmonic potential \(\psi \) on \((V,\lambda _V,J_V)\) satisfying \(\lambda _V=-\rmd \psi \circ J_V\).

The contactisation \((\R \times V,\rmd t+\lambda _V)\) of \((V,\lambda _V)\) is given by \((\R \times \C \times T^*B,\alpha _o)\) and the contact structure \(\ker \alpha _o\) is spanned by vectors of the form \(v-\lambda _V(v)\partial _t\), \(v\in TV\). Using coordinates \(s\) on the first \(\R \)-factor of \(\R \times (\R \times V)\) we define \(J\) by requiring \(J(\partial _s)=\partial _t\) and

\[ J\big (v-\lambda _V(v)\partial _t\big ) = J_Vv-\lambda _V(J_Vv)\partial _t \]

for all \(v\in TV\). In other words, \(J\) is a \(s\)-translation invariant almost complex structure on \(\R \times (\R \times V)\) that preserves the contact distributions \(\ker \alpha _o\) on all slices \(\{s\}\times \R \times V\).

By [55], Proposition 5 the Niederkrüger map

\[ \Phi (s,t,z,u)=\big (s-\psi (z,u)+\rmi t,z,u\big ) \]

is a biholomorphic map

\[ \Phi \co (\R \times \R \times \C \times T^*B,J) \lra (\C ^2\times T^*B,\rmi \oplus J_{T^*B}) \;, \]

which maps the hypersurface \(\{s=0\}\) onto \(\{s\circ \Phi ^{-1}=0\}=\{x_1=-\psi (z,u)\}\). As in [55], Proposition 3.2 we consider a \((n-1)\)-dimensional family of holomorphic discs

\[ \{-\varepsilon ^2\}\times \D _{2\varepsilon }\times \{b\} \]

in \((\C ^2\times T^*B,\rmi \oplus J_{T^*B})\) with parameters \(\varepsilon \in \R ^+\), \(b\in B\). Here, we denote by \(\D _r\subset \C \) the closed disc with radius \(r\) and centre \(0\). Writing \(\D \) for the closed unit disc \(\D _1\) the disc family can be parametrised by

\[ v_{\varepsilon ,b}(z)= \big ( -\varepsilon ^2,2\varepsilon \cdot z,b \big ) \]

for \(z\in \D \). The lifts \(u_{\varepsilon ,b}=\Phi ^{-1}\circ v_{\varepsilon ,b}\) via the Niederkrüger map are holomorphic maps

\[ (\D ,\partial \D )\lra \Big ( \R \times \R \times \C \times T^*B, \{0\}\times \{0\}\times \C ^*\times B \Big ) \]

given by

\[ u_{\varepsilon ,b}(z)= \Big ( \varepsilon ^2\big (|z|^2-1\big ),0,2\varepsilon \cdot z,b \Big ) \;. \]

We will refer to the \(u_{\varepsilon ,b}\) as local Bishop discs .

From [55], Proposition 6 we get local uniqueness :

Proof.

We remark that the boundary circles

\[ u_{\varepsilon ,b}(\partial \D )= \{0\}\times \{0\}\times \partial \D _{2\varepsilon }\times \{b\} \]

of the local Bishop discs foliate \(\{0\}\times \{0\}\times \C ^*\times B\). Recall that a neighbourhood of \(\{0\}\times \{0\}\times \{0\}\times B\) corresponds to a neighbourhood of the binding \(B\) in \(N^*\).

We consider a \(2n\)-dimensional almost complex manifold \((W,J)\) with non-empty boundary. Denote by \(M\) a boundary component \(M\subset \partial W\) and by \(\xi =TM\cap JTM\) the \(J\)-invariant hyperplane distribution along \(M\). Assume that there exists a smooth \(1\)-form \(\alpha \) on \(M\) such that \(\xi =\ker \alpha \) and that \(\rmd \alpha \) is \(J\)- positive on complex lines in \(\xi \) in the sense that \(\rmd \alpha (v,Jv)>0\) for all \(v\in \xi \), \(v\neq 0\). In other words, \((M,\xi )\) is a \(J\)- convex hypersurface of \((W,J)\) as defined in [23] .

In this situation, \(\xi \) is a contact structure with contact form \(\alpha \), cf. Remark 4.1.1 . Denoting by \(R\) the Reeb vector field of \(\alpha \) we additionally assume that \(-JR\) is outward pointing. In other words, \((M,\xi )\) is a \(J\)- convex boundary component of \((W,J)\). Observe, that \((W,J)\) is naturally oriented by the \(n\)-th power of any \(J\)-positive (and hence non-degenerate) \(2\)-form on \(W\). Therefore, the contact orientation \(\alpha \wedge (\rmd \alpha )^{n-1}\) on \(M\) and the boundary orientation on \(M\subset \partial W\) coincide.

In fact, if \(M\) is compact, then \(M\subset \partial W\) is the regular zero-level set of a strictly plurisubharmonic function \(\varrho \) defined near \(M\) that is negative on the complement of \(M\), see [19], Lemma 2.7 . The precomposition \(\varrho \circ u\) with a holomorphic map \(u\co G\ra (W,J)\), \(G\subset \C \) open domain, is subharmonic where defined, and therefore satisfies the strong maximum principle . So, for example, non-constant holomorphic spheres in \((W,J)\) are uniformly bounded away from the boundary component \(M\subset \partial W\), which we assumed to be compact. Furthermore, in the case of a non-constant \(J\)-holomorphic disc \(u\co (\D ,\partial \D )\ra (W,M)\), we get \(u(\Int \D )\subset W\setminus M\) and the radial derivative

\[ 0< \rmd \big (\varrho \circ u\big )(\partial _r) = -\big (\rmd \varrho \circ J\big ) \Big (Tu(\partial _{\theta })\Big ) \]

is positive along \(\partial \D \) by the boundary lemma of E. Hopf . Because

\[ \xi = \ker (\rmd \varrho )\cap \ker \big (\!-\rmd \varrho \circ J\big ) \]

is co-oriented by the Reeb vector field of any contact form defining \(\xi \) as a co-oriented hyperplane distribution this means that the curve \(u(\partial \D )\subset M\) is an immersion positively transverse to \(\xi \). In particular, any such holomorphic disc \(u\) such that \(u|_{\partial \D }\) is an embedding must be simple, see [27], Lemma 4.5 .

Consider a \(2n\)-dimensional almost complex manifold \((W,J)\) that has a compact \(J\)-convex boundary component \((M,\xi )\) as described in Section 3.2 . Assume that \((M,\xi )\) contains a submanifold \(N\) supporting a bordered Legendrian open book \((B,\vartheta )\). By Section 2.3 a neighbourhood of the binding \(B\subset W\) is diffeomorphic to a neighbourhood of \(\{0\}\times \{0\}\times \{0\}\times B\) in \((-\infty ,0]\times \R \times \C \times T^*B\) such that the restriction to the boundaries \(M\) and \(\{0\}\times \R \times \C \times T^*B\) induces a contactomorphism. In addition, assume that the almost complex structure \(J\) of \(W\) corresponds to the one constructed in Section 3.1 under the diffeomorphism.

From [55], Proposition 7 we get semi-global uniqueness :

Proof.

As in Section 3.1 we define an almost complex structure on \(M(\varphi ^*)\): Choose a Riemannian metric \(g_{\theta }\) on \(F\) that smoothly depends on \(\theta \in \R \) such that \(g_{2\pi }=g_{\theta }=\varphi ^*g_{\theta -2\pi }\) for all \(\theta \in (2\pi -\varepsilon ,2\pi +\varepsilon )\) and \(\varepsilon >0\) small. For each \(\theta \in \R \) we define the kinetic energy on \(T^*F\) by

\[ k_{\theta }(u)=\frac 12g_{\theta }^{\flat }(u,u) \;,\quad u\in T^*F\;, \]

denoting by \(g_{\theta }^{\flat }\) the dual metric of \(g_{\theta }\). For each \(\theta \in [0,2\pi ]\) we construct an almost complex structure \(J_{\theta }\) on \(T^*F\) as on [48], Section 5.3 that is compatible with \(\rmd \lambda _{T^*F}\) and turns \(k_{\theta }\) into a strictly plurisubharmonic potential on \(T^*F\) in the sense of [28], Section 3.1 meaning that

\[ \lambda _{T^*F}=-\rmd k_{\theta }\circ J_{\theta }\;. \]

The almost complex structure \(J_{\theta }\) is uniquely determined by \(g_{\theta }\) and \(\rmd \lambda _{T^*F}\); therefore, \(J_{\theta }\) depends smoothly on \(\theta \) and satisfies \(J_{2\pi }=J_{\theta }=\varphi ^*J_{\theta -2\pi }\) for \(\theta \in (2\pi -\varepsilon ,2\pi +\varepsilon )\).

The metric on \([0,2\pi ]\times F\) obtained by taking the sum of the Euclidean metric and \(g_{\theta }\) for each \(\theta \in [0,2\pi ]\) descents to a metric \(g\) on the mapping torus \(M(\varphi )\). The induced kinetic energy function on \(M(\varphi ^*)\) is denoted by

\[ \psi (r,\theta ,u)=\tfrac 12r^2+k_{\theta }(u) \;. \]

The almost complex structure \(\rmi \oplus J_{\theta }\) on \(T^*[0,2\pi ]\times T^*F\) descents to a compatible almost complex structure \(J_g\) on \(M(\varphi ^*)\) and

\[ r\rmd \theta +\lambda _{T^*F}\equiv -\rmd \psi \circ J_g \]

modulo second order terms in \(|u|_{\theta }\), \(u\in T^*F\), in the sense of [54], Appendix E.5 . Indeed, the derivative of \(k_{\theta }(u)\) in \(\theta \)-direction contributes a term that locally is a quadratic form in the coordinates of \(u\in T^*F\). Consequently, the restriction of \(\rmd r\wedge \rmd \theta +\rmd \lambda _{T^*F}\) to \(\{u=0\}\) in \(M(\varphi ^*)\) is equal to \(-\rmd \big (\rmd \psi \circ J_g\big )\). As \(\rmd r\wedge \rmd \theta +\rmd \lambda _{T^*F}\) is positive on \((\rmi \oplus J_{\theta })\)-complex lines and equals \(-\rmd \big (\rmd \psi \circ J_g\big )\) modulo first order terms in \(|u|_{\theta }\), \(u\in T^*F\), we conclude that \(\psi \) is strictly plurisubharmonic in a neighbourhood of \(\{u=0\}\) in \(M(\varphi ^*)\).

We consider the almost complex manifold \((\C \times M(\varphi ^*),\rmi \oplus J_g)\) provided with the Liouville form \(s\rmd t+r\rmd \theta +\lambda _{T^*F}\), where we denote the coordinates on \(\C \) by \(s+\rmi t\). Observe that the almost complex structure

\[ J=\rmi \oplus J_g \]

is compatible with \(\rmd s\wedge \rmd t+\rmd r\wedge \rmd \theta +\rmd \lambda _{T^*F}\) and that the function

\[ \Psi (s+\rmi t,r,\theta ,u)=\tfrac 12s^2+\psi (r,\theta ,u) \]

is strictly plurisubharmonic in a neighbourhood of \(\{u=0\}\) in \(\C \times M(\varphi ^*)\). This holds because

\[ s\rmd t+r\rmd \theta +\lambda _{T^*F}\equiv -\rmd \Psi \circ J \]

modulo second order terms in \(|u|_{\theta }\), \(u\in T^*F\). Therefore, as above, the restriction of \(\rmd s\wedge \rmd t+\rmd r\wedge \rmd \theta +\rmd \lambda _{T^*F}\) to \(\{u=0\}\) in \(\C \times M(\varphi ^*)\) is equal to \(-\rmd \big (\rmd \Psi \circ J\big )\).

By rescaling the metric \(g_{\theta }\) on \(F\) by a constant if necessary we can assume that \(\Psi \) is strictly plurisubharmonic in a neighbourhood of \(\{\Psi =\tfrac 12\}\). The hypersurfaces \(\{\Psi =\tfrac 12\}\) and \(\{s=1\}\) are transverse to \(s\partial _s+r\partial _r+Y_{T^*F}\), which is a Liouville vector field w.r.t. the symplectic form \(\rmd s\wedge \rmd t+\rmd r\wedge \rmd \theta +\rmd \lambda _{T^*F}\). Therefore, the contraction into the symplectic form induces contact forms on both hypersurfaces. The induced contact form on \(\{s=1\}=\{1\}\times \rmi \R \times M(\varphi ^*)\) is \(\alpha _{\varphi }\); the one on \(\{\Psi =\tfrac 12\}\) is given by \(-\rmd \Psi \circ J\) along \(\{u=0\}\).

A reparametrisation of the flow of the Liouville vector field \(s\partial _s+r\partial _r+Y_{T^*F}\) yields a contact embedding of \(\{\Psi =\tfrac 12,s>0\}\) onto \(\{s=1\}\) w.r.t. the induced contact structures, see [10], Appendix A.1 . The hypersurfaces \(\{\Psi =\tfrac 12\}\) and \(\{s=1\}\) intersect along \(\{s=1,r=0,u=0\}\), on which the flow is stationary. Moreover, as we flow along the Liouville vector field \(s\partial _s+r\partial _r+Y_{T^*F}\) we observe that the multi level set \(\{\Psi =\tfrac 12,s>0, t=0, r\leq 0, u=0\}\) corresponds to \(\{1\}\times \{0\}\times \{r\leq 0, u=0\}\) in \(\{1\}\times \rmi \R \times M(\varphi ^*)\) under the contact embedding. The latter set was used in Section 2.4 to describe the germ of contact structure near the boundary of a bordered Legendrian open book; the boundary being \(\{\Psi =\tfrac 12,s>0, t=0, r=0, u=0\}\), which corresponds to \(\{1\}\times \{0\}\times \{r=0, u=0\}\).

The hypersurface \(\{\Psi =\tfrac 12\}\) carries a second contact structure given by

\[ \ker \big (\rmd \Psi \big )\cap \ker \big (\!-\rmd \Psi \circ J\big ) \,, \]

which turns \(\{\Psi =\tfrac 12\}\) into a \(J\)-convex boundary of \(\{\Psi \leq \tfrac 12\}\). The induced singular characteristic foliation on \(\{\Psi =\tfrac 12,s>0, t=0, r\leq 0, u=0\}\) coincides with the one described in the preceding paragraph, where the contact distribution this time is taken w.r.t. \(s\rmd t+r\rmd \theta +\lambda _{T^*F}\equiv -\rmd \Psi \circ J\) along \(\{u=0\}\). By uniqueness of the germ of a contact structure formulated in [50], Lemma 4.6 , a neighbourhood of \(\{\Psi =\tfrac 12,s>0, t=0, r\leq 0, u=0\}\) is contactomorphic to the contact structure we considered first. In other words, given a contact manifold \((M,\xi )\) containing a submanifold \(N\) that supports a bordered Legendrian open book, we obtain an alternative description of the germ of contact structure near the boundary \(\partial N\) presented in Section 2.4 .

Let \((W,J)\) be a \(2n\)-dimensional almost complex manifold so that a given contact manifold \((M,\xi )\) is a compact \(J\)-convex boundary component of \(W\), see Section 3.2 . Let \(N\) be a submanifold of \((M,\xi )\) that supports a bordered Legendrian open book \((B,\vartheta )\). In view of Section 3.4 we assume that a neighbourhood of the boundary \(\partial N\subset W\) is diffeomorphic to a neighbourhood of \(\{1\}\times \{0\}\times \{r=0, u=0\}\) in \(\{\Psi \leq \tfrac 12, s>0\}\) such that \(N\subset W\) and \(\{\Psi =\tfrac 12,s>0, t=0, r\leq 0, u=0\}\) correspond to each other diffeomorphically. Furthermore we assume that under the diffeomorphism the almost complex structure \(J\) of \(W\) corresponds to \(\rmi \oplus J_g\) on \(\{\Psi \leq \tfrac 12\}\) in \(\C \times M(\varphi ^*)\) inducing contactomorphisms on the boundary.

Similarly to [50], Lemma 4.7 we obtain the blocking lemma :

Proof.

Let \(\alpha \) be a defining contact form for \(\xi \) on \(M\) such that \(\rmd \alpha \) is positive on complex lines in \((\xi ,J)\).

The symplectic neighbourhood theorem for hypersurfaces implies that there exist \(\varepsilon >0\) and a symplectic embedding

\[ \Big ((-\varepsilon ,0]\times M,\rmd (s\alpha )+\omega \Big ) \lra (W,\Omega ) \]

whose restriction to \(\{0\}\times M\) is the inclusion of \(M=\partial W\) into \(W\), cf. [52], Exercise 3.36 and [50], Remark 2.7 . Observe that

\[ \Big (\rmd (s\alpha )+\omega \Big )^n= \Big (\rmd s\wedge \alpha +s\rmd \alpha +\omega \Big )^n= n\rmd s\wedge \alpha \wedge \big (s\rmd \alpha +\omega \big )^{n-1} \,. \]

Therefore, we find \(\varepsilon >0\) and a large positive constant \(s_o\) such that, considered on \((-\varepsilon ,\infty )\times M\), \(\rmd (s\alpha )+\omega \) is symplectic on \((-\varepsilon ,\varepsilon )\times M\) and \((s_o,\infty )\times M\). In fact, using Remark 4.1.1 , \(\alpha \wedge \big (s\rmd \alpha +\omega \big )^{n-1}\) is a positive volume form on \(M\) for all positive \(s\) because \(\rmd \alpha \) and \(\omega \) are positive on complex lines in \((\xi ,J)\). Therefore, \(\rmd (s\alpha )+\omega \) is a symplectic form on \((-\varepsilon ,\infty )\times M\). Via gluing along \(\{0\}\times M\equiv M\subset \partial W\) using the above symplectic embedding we build a symplectic manifold, the so-called magnetic completion ,

\[ \big (\widetilde {W},\widetilde {\Omega }\big ):= (W,\Omega )\cup \Big ([0,\infty )\times M,\rmd (s\alpha )+\omega \Big ) \,. \]

We continue the considerations in Section 4.1 . In addition, let \(N\subset (M,\xi )\) be a submanifold that carries the structure of a bordered Legendrian open book decomposition \((B,\vartheta )\). Choose contact embeddings of the model neighbourhood of the binding \(B\) (see Section 2.3 ) and of the alternative model neighbourhood of the boundary \(\partial N\) (see Section 3.4 ) into \((M,\xi =\ker \alpha )\). The push forward of the respective contact forms \(\alpha _o\) and the restriction of \(-\rmd \Psi \circ J\) to the tangent spaces of \(\{\Psi =\tfrac 12\}\) are equal to \(\rme ^h\alpha \) for a smooth function \(h\) on the images of the model neighbourhoods. Alter the contact form \(\alpha \) by cutting off \(h\) to \(0\) near the boundary of these images, so that the considered contact embeddings – after shrinking the model neighbourhoods a bit – are in fact strict. Observe that this conformal change of the contact form \(\alpha \) does not effect the property of \(\rmd \alpha \) to be positive on complex lines in \((\xi ,J)\).

Let \(s_o\) be a positive real number and consider the truncation

\[ \big (\hat {W},\hat {\Omega }\big ):= (W,\Omega )\cup \Big ([0,2s_o]\times M,\rmd (s\alpha )+\omega \Big ) \;. \]

The resulting contact embeddings of the model neighbourhoods into \(\{2s_o\}\times M\), which are strict up to conformal factor \(2s_o\) w.r.t. the contact form \(2s_o\alpha \), extend along collar directions to embeddings of the neighbourhood \(U_B\) constructed in Lemma 3.3.1 and of the neighbourhood \(U_{\partial N}\) constructed in Lemma 3.5.1 into \((-\infty ,2s_o]\times M\) in the following way: For the embedding of \(U_B\) simply cross with the identity in the \(s\)-direction; for the embedding of \(U_{\partial N}\) denote by \(Y\) the vector field on \(\{\Psi \leq \tfrac 12\}\) obtained by multiplying the Reeb vector field of the contact form \(-\rmd \Psi \circ J\) on the level sets of \(\Psi \) with \(-J\) and follow the flow lines of \(Y\) in backward time (taking time logarithmically). Observe that \(Y\) coincides with the Liouville vector field \(s\partial _s+r\partial _r+Y_{T^*F}\) along \(\{u=0\}\), see Section 3.4 .

The images of the neighbourhoods are again denoted by \(U_B\) and \(U_{\partial N}\). Taking \(s_o\) sufficiently large we achieve that \(U_B\cup U_{\partial N}\) fit into \([s_o,2s_o]\times M\). Push forward yields an almost complex structure \(J_o\) on \(U_B\cup U_{\partial N}\) that allows the conclusions of Lemmata 3.3.1 and 3.5.1 . We remark that \(J_o\) is invariant under translations in \(s\)-direction that shift off \(U_B\); along \([s_o,2s_o]\times \partial N\) the almost complex structure \(J_o\) is independent of \(s\).

Moreover, the symplectic form \(\rmd (s\alpha )\) is compatible with \(J_o\) on \(U_B\) and on \(U_{\partial N}\), see Remark 3.1.1 and Section 3.4 , resp. Furthermore \(J_o\) sends \(\partial _s\) to the Reeb vector field and preserves the contact distribution. Applying [52], Proposition 2.63(i) to the symplectic bundle

\[ \big (\xi ,\rmd (s\alpha )\big )\lra [s_o,2s_o]\times M \]

we extend \(J_o\) to an almost complex structure on \([s_o,2s_o]\times M\) keeping the properties just listed. In particular, \(\{2s_o\}\times M\) is a \(J_o\)-convex boundary independently of the conformal factor \(2s_o\), see Section 3.2 .

Placing the whole scenario to \([s_o,2s_o]\times M\) for \(s_o\) sufficiently large we can additionally assume that \(J_o\) is also tamed by the symplectic form \(\rmd (s\alpha )+\omega \) on \([s_o,2s_o]\times M\). This is possible because for \(s_o\) sufficiently large \(\rmd (s\alpha )=\rmd s\wedge \alpha +s\rmd \alpha \) dominates \(\rmd (s\alpha )+\omega =\rmd s\wedge \alpha +(s\rmd \alpha +\omega )\) on \(J_o\)-complex lines. With [52], Proposition 2.51 we extend \(J_o\) to a tamed almost complex structure \(\hat {J}\) on \(\big (\hat {W},\hat {\Omega }\big )\) that restricts to \(J\) on \(W\) and to \(J_o\) on \([s_o,2s_o]\times M\). Moreover, \(\{2s_o\}\times M\subset \partial \hat {W}\) is a \(\hat {J}\)-convex boundary component. We will refer to the construction of \(\big (\hat {W},\hat {\Omega },\hat {J}\big )\) as magnetic collar extension .

Assuming exactness of \(\omega |_{TN}\) in the situation of Sections 4.1 and 4.2 the symplectic form in the magnetic collar extension \(\big (\hat {W},\hat {\Omega },\hat {J}\big )\) can be assumed to satisfy

\[ \hat {\Omega }=2s_o\rmd \alpha \qquad \text {on}\quad T\big (\{2s_o\}\times N\big ) \]

after deformation:

Write \(\omega =\rmd \beta \) in a neighbourhood of \(N\) taking a neighbourhood that strongly deformation retracts to \(N\). Extend \(\beta \) to a \(1\)-form on \(M\) that vanishes outside a larger neighbourhood. Define a \(2\)-form \(\eta :=\omega -\rmd (\varrho \beta )\) on \([s_o,2s_o]\times M\), where \(\varrho \) is a smooth function that vanishes on \(\{s\leq s_o\}\), equals \(1\) on \(\{s\geq 2s_o\}\), and satisfies \(0\leq \varrho '\leq 2/s_o\). We claim that for \(s_o\) sufficiently large

\[ \Big ([s_o,2s_o]\times M,\rmd (s\alpha )+\eta \Big ) \]

is symplectic. Indeed, spelling out the \(n\)-th power of \(\rmd (s\alpha )+\eta \) we find

\[ n\rmd s\wedge \big (\alpha -\varrho '\beta \big ) \wedge \Big (s\rmd \alpha +\omega -\varrho \rmd \beta \Big )^{n-1} \,. \]

For \(s_o\) sufficiently large \(\alpha -\varrho '\beta \) will be a contact form on all slices \(\{s\}\times M\), \(s\in [s_o,2s_o]\), so that, restricted to the related contact structures, \((s\rmd \alpha )^{n-1}\) is a positive volume form. Making \(s_o\) even larger \((s\rmd \alpha )^{n-1}\) dominates lower order terms in the \((n-1)\)-st power of \(s\rmd \alpha +\omega -\varrho \rmd \beta \) restricted to the mentioned contact structures.

Starting with the deformed symplectic structure corresponding to \(\rmd (s\alpha )+\eta \) the tamed complex structure \(\hat {J}\) can be constructed as in Section 4.2 . In order to achieve that \(\rmd (s\alpha )\) dominates

\[ \rmd (s\alpha )+\eta = \rmd s\wedge \big (\alpha -\varrho '\beta \big ) +\Big (s\rmd \alpha +\omega -\varrho \rmd \beta \Big ) \]

on \(J_o\)-complex lines on \([s_o,2s_o]\times M\) simply choose \(s_o\) sufficiently large.

We will refer to the construction of \(\big (\hat {W},\hat {\Omega },2s_o\alpha ,\hat {J}\big )\) as deformed magnetic collar extension .

We consider a deformed magnetic collar extension of the \(\Omega \)-tamed almost complex structure \((W,J)\) as in Section 4.3 . The resulting tamed almost complex manifold together with the choice of contact form on the \(J\)-convex boundary component is denoted by \((W,\Omega ,\alpha ,J)\). In particular, the restriction of the \(2\)-forms \(\Omega \) and \(\rmd \alpha \) to the tangent spaces of the bordered Legendrian open book \((N,B,\vartheta )\subset (M,\xi =\ker \alpha )\) are equal as \(2\)-form on \(N\).

Notice, that the regular set \(N^*\) of \(N\) (see Section 2.1 ) is a totally real submanifold of \((W,J)\), i.e. \(TN^*\cap JTN^*\) is the zero section. Indeed, denoting by \(E\) the real linear span of \(v,Jv\) for a given tangent vector \(v\in T_pN^*\) the only possibility for the complex line \(E\) to be tangent to \(N^*\subset M\) is to be tangent to the page of \((B,\vartheta )\) through \(p\), which is Legendrian w.r.t. \(\xi =TM\cap JTM\). Positivity of \(\rmd \alpha \) on complex lines in \((\xi ,J)\) implies the vanishing of \(v\) because the pages of \((B,\vartheta )\) are Legendrian submanifolds, and, hence, have Lagrangian tangent spaces inside \((\xi ,\rmd \alpha )\).

Via the neighbourhood \(U_B\subset W\) of the binding \(B\subset N\) constructed in Lemma 3.3.1 we obtain an embedding relative boundary, a so-called local Bishop filling ,

\[ F\co \Big ( (0,\delta )\times \D \times B, (0,\delta )\times \partial \D \times B \Big )\lra (W,N^*) \]

for some \(\delta >0\) such that for all \((\varepsilon ,b)\in (0,\delta )\times B\) the maps \(u_{\varepsilon ,b}=F(\varepsilon ,\,.\,,b)\) are \(J\)-holomorphic discs \((\D ,\partial \D )\ra (W,N^*)\). Furthermore \(F\) extends smoothly to a map defined on \([0,\delta )\times \D \times B\) that maps \(\{0\}\times \D \times B\) to \(B\) via \((\varepsilon ,z,b)\mapsto b\).

We consider the moduli space \(\MM \) of \(J\)-holomorphic discs \(u\co (\D ,\partial \D )\ra (W,N^*)\) that are homologous in \(W\) relative \(N^*\) to one of the Bishop discs \(u_{\varepsilon ,b}\). For all \(u\in \MM \) we have the following:

1. The degree of the map \(\vartheta \circ u\co \partial \D \ra S^1\), the so-called winding number of \(u\), is equal to \(1\). The boundary lemma of E. Hopf implies, that \(u(\partial \D )\) is an embedded curve in \(N^*\) positively transverse to \(\xi \). Hence, \(u\) is simple, see Section 3.2 .
2. We find a \(2\)-chain \(C\) in \(N\) with boundary \(-u(\partial \D )\) such \(u(\D )+C\) is null-homologous in \(W\). Therefore, by applying Stokes theorem twice, we get that the symplectic energy of \(u\) satisfies

   \[ 0<\int _{\D }u^*\Omega =-\int _{C}\Omega =\int _{\partial \D }u^*\alpha \,, \]

   as \(\Omega =\rmd \alpha \) on \(TN\). Denote by \(\rmd \vartheta \) the pullback of \(\rmd \theta \) along \(\vartheta \co N\setminus B\ra S^1\). According to our co-orientation convention in Section 2.1 and the local models in Sections 2.3 and 2.4 we observe that \(\alpha |_{TN}=f\rmd \vartheta \) for a non-negative function \(f\) on \(N\), that vanishes precisely along the singular set \(B\cup \partial N\). Consequently, we get uniform energy bounds

   \[ 0<\int _{\D }u^*\Omega = \int _{\partial \D }u^*f\cdot (\vartheta \circ u)^*\rmd \theta \leq 2\pi \max f\,. \]

3. The restriction of \(u\) to \(\partial \D \) is uniformly bounded away from the boundary \(\partial N\) as the intersection of \(u(\D )\) with \(U_{\partial N}\) is empty by Lemma 3.5.1 . Recall that if \(u(\D )\) intersects the neighbourhood \(U_B\) of the binding then \(u\) is a reparametrisation of a local Bishop disc \(u_{\varepsilon ,b}\), see Lemma 3.3.1 . We truncate the moduli space \(\MM \) (keeping the notation) by removing all holomorphic discs that are reparametrisations of \(u_{\varepsilon ,b}\) for \((\varepsilon ,b)\in (0,\delta /2)\times B\).

We consider a symplectic cobordism \((W,\Omega )\). We assume that the boundary \(\partial (W,\Omega )\) decomposes into concave boundary components, whose union we denote by \((M_-,\omega _-,\alpha _-)\), and positive boundary components, whose union we denote by \((M_+,\omega _+,\alpha _+)\). In addition, we require the weak-filling condition : \(\alpha _+\) can be chosen to be a contact form, which according to our conventions implies that \(\alpha _+\wedge (\rmd \alpha _+)^{n-1}\) and \(\alpha _+\wedge \omega _+^{n-1}\) are positive, such that

\[ \alpha _+\wedge \big (f_+\rmd \alpha _++\omega _+\big )^{n-1} >0 \]

for all non-negative smooth functions \(f_+\) on \(M_+\). We call \((W,\Omega )\) a directed symplectic cobordism , cf. [24] .

As in [28] , [64] such symplectic cobordisms \((W,\Omega )\) can be used to verify the strong Weinstein conjecture for the contact manifold that appears as the concave boundary component of \((W,\Omega )\). The \(3\)-dimensional variant is related to the ball theorem, see [29], Theorem 2.2 and Corollary 3.8 :

Theorem 5.1.2 part (i) is contained in [50] . We include a variant of the argument in Section 5.3 as guideline for the polyfold proof of part (ii). Following [54] we call a \(2n\)-dimensional symplectic manifold \((W,\Omega )\) semi-positive if the first Chern class \(c_1\) of \((W,\Omega )\) satisfies the following condition: \(c_1(A)\geq 0\) for all spherical homology \(2\)-classes \(A\) of \(W\) with \(c_1(A)\geq 3-n\) and \(\Omega (A)>0\). If \(2n\leq 6\) the condition is automatic. The proof of part (ii) of Theorem 5.1.2 is postponed to Section 7 .

We remark that the inclusion \(N^*\subset W\) induces in cohomology the restriction map \(H^2(W;\Z _2)\ra H^2(N^*;\Z _2)\). The long exact cohomology sequence with \(\Z _2\)-coefficients yields, that the restriction map is surjective if and only if the co-boundary operator \(H^2(N^*;\Z _2)\ra H^3(W,N^*;\Z _2)\) vanishes. Furthermore the choice of an orientation on \(N^*\) induces an orientation on each page \(\vartheta ^{-1}(\theta )\), \(\theta \in S^1\), via the co-orientation induced by \(\vartheta \). The binding is oriented via the boundary orientation of a page. Therefore, \(N\) is orientable if and only if \(N^*\) is.

Observe that the inclusion map \(f\co N^*\subset N\) induces \(TN^*=f^*TN\), so that naturality of the Stiefel–Whitney classes implies \(w_2(TN^*)=f^*w_2(TN)\). In particular, if \(w_2(TN)\) vanishes so does \(w_2(TN^*)\). Further, if the fibration \(\vartheta \) on \(N^*\) is trivial (as it is the case for the plastikstufe), e.g. a product of the page \(\vartheta ^{-1}(1)\) with \(S^1\), the second Stiefel–Whitney class \(w_2(TN^*)\) is equal to \(w_2\big (T\vartheta ^{-1}(1)\big )\) according to the Whitney cross product formula for the total Stiefel–Whitney class, cf. [16], Chapter 17 . In general, taking a connection on the fibration \(\vartheta \) we obtain a splitting of \(TN^*\) into the vertical \(\ker T\vartheta \) and horizontal subbundle. As the horizontal subbundle is isomorphic to the trivial bundle \(\vartheta ^*TS^1\) Whitney sum formula for the total Stiefel–Whitney class implies \(w_2(TN^*)=w_2\big (\!\ker T\vartheta \big )\). The square product of the \(2\)-dimensional Klein bottle shows that the latter not always equals the second Stiefel–Whitney class of a fibre.

We consider the directed symplectic cobordism \((W,\Omega )\) from Theorem 5.1.2 . According to the contact type boundary condition along the negative boundary components of \((W,\Omega )\) the symplectic neighbourhood theorem allows a description of a collar neighbourhood of \(M_-\subset W\) as \([0,\varepsilon )\times M_-\) such that the symplectic form equals \(\Omega =\rmd (\rme ^s\alpha _-)\) with \(s\in [0,\varepsilon )\). Therefore, a partial completion over the concave boundary of \((W,\Omega )\) can be given by

\[ \Big ((-\infty ,0]\times M_-,\rmd (\rme ^s\alpha _-)\Big ) \cup (W,\Omega ) \]

via gluing along \(\{0\}\times M_-\equiv M_-\). Any other contact form defining \(\xi _-=\ker \alpha _-\) on \(M_-\) can be realised as the restriction to the tangent spaces of a graph over \(M_-\) inside \((-\infty ,\varepsilon )\times M_-\) up to a positive constant factor, cf. [28], Section 3.3 . A change of \((W,\Omega )\) by adding the super-level set of the graph to the directed symplectic cobordism will allow the verification of the strong Weinstein conjecture for a particular choice of contact form as announced in Theorem 5.1.2 . In fact, we will assume that \(\alpha _-\) is a generic perturbation of a given \(\xi _-\)-defining contact form which allows an application of the Gromov–Hofer compactness theorem as formulated in [15] . This is justifiable with the Arzelà–Ascoli theorem, cf. [28], Section 6.4 and [64], Section 6 .

On the negative end \(\big ((-\infty ,0]\times M_-,\rmd (\rme ^s\alpha _-)\big )\) we choose a shift invariant almost complex structure \(J_-\) that sends the Liouville vector field \(\partial _s\) to the Reeb vector field of \(\alpha _-\) and leaves \(\xi _-\) invariant such that \(J_-\) is compatible with the symplectic structure induced by \(\rmd \alpha _-\) on \(\xi _-\). Extend \(J_-\) to a tamed almost complex structure \(J\) on \((W,\Omega )\) such that \(\xi _+=TM_+\cap JTM_+\) and the positive boundary \((M_+,\xi _+)\) of \((W,\Omega )\) is \(J\)-convex, see [50], Theorem D . Further, we glue the deformed magnetic collar extension constructed in Sections 4.2 and 4.3 along \(M_+\equiv \{0\}\times M_+\) to build (keeping the notation) a symplectic manifold

\[ \big (\hat {W},\hat {\Omega }\big ):= \Big ((-\infty ,0]\times M_-,\rmd (\rme ^s\alpha _-)\Big ) \cup (W,\Omega )\cup \Big ([0,2s_o]\times M_+,\rmd (s\alpha _+)+\eta \Big ) \,. \]

The resulting almost complex structure is denoted by \(\hat {J}\). Notice that \(\hat {J}\) equals \(J_o\) on the neighbourhoods \(U_B\) and \(U_{\partial N}\) of \(B\) and \(\partial N\), resp. Here we think of \(B\) and \(\partial N\) as subsets of \(M_+\equiv \{2s_o\}\times M_+\). In particular, the results from Section 4.4 are available.

We prove Theorem 5.1.2 under assumption (i). Recall the moduli space \(\MM \) introduced in Section 4.4 . To cut out the Möbius reparametrisation group geometrically we define the moduli space \(\MM _{1,\rmi ,-1}\) to be the set of all holomorphic discs \(u\in \MM \) such that \(\vartheta \circ u(\rmi ^k)=\rmi ^k\) for \(k=0,1,2\). This allows to fix the disc reparametrisations for sequences in \(\MM _{1,\rmi ,-1}\) in the compactness formulation in Remark 4.4.1 .

We choose a base point \(b_o\) of \(B\) and an embedded curve \(\gamma \) inside the page \(\vartheta ^{-1}(1)\) that connects \(B\) and \(\partial N\) such that \(\gamma \) is given by \(\{b_o\}\times \big (D^2\cap \R _{\geq 0}\big )\) in the model description in Section 2.3 . We define the moduli space \(\MM _{\gamma }\) to be the set of all holomorphic discs \(u\in \MM _{1,\rmi ,-1}\) with \(u(1)\in \gamma \). In other words \(\MM _{\gamma }\) consists of all \(u\in \MM \) such that

\[ u(1)\in \gamma \,,\quad \vartheta \circ u(\rmi )=\rmi \,,\quad \vartheta \circ u(-1)=-1 \,. \]

The Maslov index of the Fredholm problem defined by \(\MM \) equals \(2\) (see [55], Proposition 8 or [27] ). With [68], Theorem 4.4 there is a generic choice of \(\hat {J}\) such that with the first dimension formula in [68], Theorem 3.7 (successively taking relations \(R=\emptyset \), \(R=\vartheta ^{-1}(1)\times \vartheta ^{-1}(\rmi )\times \vartheta ^{-1}(-1)\), and \(R=\gamma \times \vartheta ^{-1}(\rmi )\times \vartheta ^{-1}(-1)\)) the moduli spaces \(\MM \), \(\MM _{1,\rmi ,-1}\), and \(\MM _{\gamma }\) are smooth manifolds of dimension \(n+2\), \(n-1\), and \(1\), resp. By [68], Remark 3.6 we can assume that the generic perturbation of \(\hat {J}\) is supported in the complement of the union of the negative end \((-\infty ,0]\times M_-\) of \(\hat {W}\) and \(U_B\cup U_{\partial N}\). This is because all local Bishop discs are Fredholm regular by [55], Proposition 9 and because all holomorphic discs that are contained completely inside \(\big ((-\infty ,0]\times M_-\big )\cup U_B\cup U_{\partial N}\) are the local Bishop discs.

Moreover, the boundary component of \(\MM _{1,\rmi ,-1}\) that corresponds to the local Bishop filling \(F\) has a collar neighbourhood in \(\MM _{1,\rmi ,-1}\) diffeomorphic to \([0,1)\times B\) via \(F\). This results in a collar neighbourhood \([0,1)\times \{b_o\}\) in \(\MM _{\gamma }\), see Section 4.4 . By the blocking property of \(U_{\partial N}\) (see Section 3.5 ) the evaluation map \(\MM _{\gamma }\ra \gamma \), \(u\mapsto u(1)\), is not surjective. Invariance of the mod-\(2\) degree for proper maps, which counts the number of preimages modulo \(2\), implies that \(\MM _{\gamma }\) cannot be compact.

Proof of Theorem 5.1.2 part (i) .

Let \(S\) be an oriented surface that is equal to the disjoint union of one closed disc and a (possibly empty) finite collection of spheres. All connected components of \(S\) are provided with the standard orientation. Let \(j\) be an orientation preserving complex structure on \(S\) turning \((S,j)\) into a Riemann surface with boundary, i.e. \((S,j)\) admits a holomorphic atlas whose charts are given by open subsets of the closed upper half-plane.

We call a subset of \(\Int (S)\) consisting of two distinct points a nodal pair . Each finite collection \(D\) of pair-wise disjoint nodal pairs defines an equivalence relation on \(S\) calling two points equivalent if and only if they from a nodal pair. The set \(S/D\) of equivalence classes is provided with the quotient space topology.

Let \(D\) be a finite collection of pair-wise disjoint nodal pairs such the quotient space \(S/D\) is simply connected. We call \((S,j,D)\) a boundary un-noded nodal disc . A point of \(S\) that belongs to a nodal pair in \(D\) is called a nodal point . The set of nodal points is denoted by \(|D|\). Observe that \(|D|\) and \(\partial S\) are disjoint.

Let \(m_0,m_1,m_2\) be pair-wise distinct marked points on the boundary \(\partial S\) ordered according to the boundary orientation of \(\partial S\). We call \(\big (S,j,D,\{m_0,m_1,m_2\}\big )\) a marked boundary un-noded nodal disc . Two marked boundary un-noded nodal discs

\[ \big (S,j,D,\{m_0,m_1,m_2\}\big ) \quad \text {and} \quad \big (S',j',D',\{m'_0,m'_1,m'_2\}\big ) \]

are equivalent if there exists a diffeomorphism \(\varphi \co S\ra S'\) such that \(\varphi ^*j'=j\), the injection \(D\ra D'\) defined by \(\{\varphi (x),\varphi (y)\}\in D'\) for all \(\{x,y\}\in D\) is onto, and \(\varphi (m_k)=m'_k\) for \(k=0,1,2\). Observe that \(\varphi \) necessarily preserves orientations.

In Section 7.2 boundary un-noded nodal discs will appear as the domain of stable maps. If the domain nodal discs have sphere components, the nodal discs will be unstable. In order to obtain a natural groupoidal structure on the space of marked boundary un-noded nodal discs we have to stabilise these by adding marked points. A point that is a marked point or a nodal point is called special . We call a connected component \(C\) of \(S\) stable if the number of special points on \(C\) is greater or equal than \(3\). In particular the disc component of \(S\) is stable.

We consider equivalence classes of stable discs \(\big [S,j,D,\{m_0,m_1,m_2\},A\big ]\) where \(\big (S,j,D,\{m_0,m_1,m_2\}\big )\) is a marked boundary un-noded nodal disc as in Section 6.1 that we provide with an additional finite set of auxiliary marked points \(A\subset S\setminus \partial S\) in the complement of \(|D|\) so that \(\#\big ((A\cup |D|)\cap C\big )\geq 3\) for each sphere component \(C\) of \(S\). In particular, all components \(C\) of \(S\) are stable. The equivalence relation is given by diffeomorphisms \(\varphi \co S\ra S'\) as in Section 6.1 such that in addition \(\varphi \) maps \(A\) bijectively onto \(A'\).

The set of all equivalence classes \(\RR \) is a nodal Riemann moduli space . Given a non-negative integer \(N\) we denote by \(\RR _N\subset \RR \) the subset of stable nodal discs that are equipped with precisely \(N=\#A\) auxiliary marked points so that \(\RR \) is the disjoint union over all \(\RR _N\), \(N\geq 0\). The elements in \(\RR _N\) can be represented by stable nodal discs \(\big (S,j,D,\{m_0,m_1,m_2\},A\big )\) of different stable nodal type \(\tau \), which is an isomorphism class of weighted rooted trees. The vertices are given by the components of \(S\), where the root corresponds to the disc component. All vertices are weighted by the number of (auxiliary) marked points on the corresponding component of \(S\). The edge relation is given by the nodes in \(D\). Observe that for given \(N\) the number of stable nodal types corresponding to stable discs \(\big (S,j,D,\{m_0,m_1,m_2\},A\big )\) with \(N=\#A\) is finite so that \(\RR _N\) is a finite disjoint union of subsets of stable discs \(\RR _{\tau }\subset \RR \) of the same stable nodal type \(\tau \).

Let \(\tau \) be a stable nodal type. In order to rewrite \(\RR _{\tau }\) as an orbit space we choose natural representatives of the stable nodal marked discs \(\big [S,j,D,\{m_0,m_1,m_2\},A\big ]\) in \(\RR _{\tau }\) as follows: We fix the oriented diffeomorphism types of \(\C P^1\) and \(\D \) for the components of \(S\) including the choice \(\{1,\rmi ,-1\}\) for the marked points \(\{m_0,m_1,m_2\}\), i.e. \(m_k=\rmi ^k\), \(k=0,1,2\). We denote by \(\sigma \) the area form on \(S\) that is the Fubini–Study form on the \(\C P^1\) components and equals \(\rmd x\wedge \rmd y\) on \(\D \) taking conformal coordinates \(x+\rmi y\). Let \(\JJ \equiv \JJ _S\) be the space of orientation preserving complex structures on \(S\), which equals the space of almost complex structures on \(S\) tamed by \(\sigma \), cf. [1] , [63] . Furthermore we fix a selection of special points \(|D|\) and \(A\). By \(\GG \equiv \GG (S,D,\{1,\rmi ,-1\},A)\) we denote the group of orientation preserving diffeomorphisms of \(S\) that preserve \(\{1,\rmi ,-1\}\) point-wise and inject \(A\) onto \(A\) and \(D\) onto \(D\), resp.

We identify the nodal Riemann moduli space \(\RR _{\tau }\) with the orbit space

\[ \RR _{\tau }=\JJ /\GG \, \qquad \text {via}\quad \big [S,j,D,\{1,\rmi ,-1\},A\big ]\equiv [j]\,, \]

cf. [61], p. 612 or [67], Section 4.2 . The action

\[ \GG \times \JJ \lra \JJ \,,\quad (\varphi ,j)\longmapsto \varphi ^*j\,, \]

of \(\GG \) on \(\JJ \) is given by the pull back

\[ \varphi ^*j:=T_{\varphi }\varphi ^{-1}\circ j_{\varphi }\circ T\varphi \]

for \(\varphi \in \GG \) and \(j\in \JJ \). By [61], Lemma 7.5 or [67], Lemma 4.2.8 this action is proper, see also Remark 6.3.2 below. The action is free if and only if all isotropy subgroups \(\GG _j\), \(j\in \JJ \), of \(j\)-holomorphic maps in \(\GG \) are trivial. The action is locally free because all isotropy subgroups \(\GG _j\) are finite by the following Remark 6.3.1 :

We assume the situation of Section 6.3 . Denote by \(\Omega ^0\) the Lie algebra given by the tangent space of \(\GG \) at the identity diffeomorphism, which is the space of vector fields on \((S,\partial S)\) that are stationary on the set of all special points. In other words, \(\Omega ^0\) is the set of all smooth sections of \(TS\) that are tangent to \(\partial S\) and vanish on \(|D|\cup \{1,\rmi ,-1\}\cup A\). We denote by \(\Omega ^{0,1}_j\) the space of all endomorphism fields of the tangent bundle of \(S\) that anti-commute with \(j\). Linearising the equation \(j^2=-1\) we see that \(\Omega ^{0,1}_j\) is the tangent space of \(\JJ \) at \(j\). Viewing \(\Omega ^{0,1}_j\) as the set of all \(j\)-complex anti-linear \(TS\)-valued differential forms on \(S\), the infinitesimal action is

\[ \Omega ^0\times \Omega ^{0,1}_j\lra \Omega ^{0,1}_j \,,\quad (X,y)\longmapsto L_Xj+y \,. \]

A complex linear Cauchy–Riemann operator \(D\) is a \(j\)-complex linear operator that maps smooth vector fields of \((S,j)\) to \(\Omega ^{0,1}_j\) such that \(D(fX)=\DB f\cdot X+fDX\) for all smooth vector fields \(X\) and smooth functions \(f\), where \(\DB \) is the composition of the exterior derivative with the projection of the space of smooth \(1\)-forms onto those that anti-commute with \(j\), see [54], Appendix C.1 . Complex linearity can be expressed via \(D(jX)=jDX\) for all vector fields \(X\) of \(S\).

In order to compute the Lie derivative \(L_Xj\) we denote by \(\CR j\) the uniquely determined complex linear Cauchy–Riemann operator of the holomorphic line bundle \((TS,j)\) (ignoring boundary points) that agrees with

\[ \DB X=\tfrac 12\big (TX+\rmi \circ TX\circ \rmi \big ) \]

in local holomorphic coordinates \((\C ,\rmi )\), cf. [54], Remark C.1.1 . The local holomorphic representation \(\DB \) of \(\CR j\) implies that \(\CR j\) induces a real linear Cauchy–Riemann operator on the bundle pair \(\big ((TS,T\partial S),j\big )\) taking local holomorphic coordinates in the closed upper half-plane. Nevertheless the operator \(\CR j\co \Omega ^0\ra \Omega ^{0,1}_j\) is not complex linear because \(j\) does not induce a complex linear vector space structure on \(\Omega ^0\). Indeed, \(jX\) is not tangent to \(\partial S\) for all boundary points of \(S\) for which \(X\in \Omega ^0\) does not vanish.

With this preparation we compute

\[ L_Xj=\frac {\rmd }{\rmd t}\Big |_{t=0}\varphi _t^*j\,, \]

where \(\varphi _t\) is a smooth path in \(\GG \) through \(\varphi _0=\id \) with

\[ \frac {\rmd }{\rmd t}\Big |_{t=0}\varphi _t=X\,. \]

In local holomorphic coordinates \(\varphi _t^*j\) reads as

\[ T_{\varphi _t}\varphi _t^{-1}\circ \rmi \circ T\varphi _t\,. \]

Taking the time derivative at \(t=0\) and using \(\varphi _t^{-1}=\varphi _{-t}\) yields

\[ -TX\circ \rmi +\rmi \circ TX= \rmi \big (\rmi \circ TX\circ \rmi +TX\big )\,, \]

so that the Lie derivative is equal to

\[ L_Xj=2j\CR jX \,. \]

We compute the Fredholm index of the \(W^{1,3}\)-Sobolev completed Cauchy–Riemann operator \(\CR j\co W^{1,3}\ra L^3\) induced by the Cauchy–Riemann operator from Section 6.4 .

Ignoring zeros for the moment, for each component \(C\) of \(S\) the Fredholm index is given by the Riemann–Roch [54], Theorem C.1.10 applied to the \(j\)-complex \(1\)-dimensional bundle pair \((TC,T\partial C)\). Namely, the Fredholm index is the sum of the Euler characteristic of \(C\) and the Maslov index of \((TC,T\partial C)\). The Maslov index for \(C=\D \) is \(2\) by normalisation; for \(C=\C P^1\) twice the first Chern number, i.e. twice the Euler characteristic, which gives \(4\), see [54], Chapter C.3 . Hence, the Fredholm index is \(3\) restricted to the disc component and \(6\) on the spheres.

Now we take the zeros into account. Component-wise, for each boundary zero we have to subtract \(1\) from the computed Fredholm index; for each interior zero we subtract \(2\). Adding up, we obtain that \(\ind \CR j\) equals

\[ 3-\#\{1,\rmi ,-1\}-2\#\big ((|D|\cup A)\cap \D \big ) +2\sum _C\Big (3-\#\big ((|D|\cup A)\cap C\big )\Big ) \,, \]

where the sum is taken over all sphere components \(C\) of \(S\). This gives

\[ \ind \CR j= -2\#|D|-2\#A +6\big (\#\{C\}-1\big ) \,, \]

where \(\#\{C\}=\#D+1\) is the number of all components of \(S\). Using \(\#|D|=2\#D\) we finally obtain

\[ \ind \CR j=2\big (\#D-\#A\big ) \,. \]

On the other hand, by a boundary version of the argument principle (see [3], Theorem A.5.4 ), the Maslov index of \((TC,T\partial C)\) for each component \(C\) of \(S\) is the weighted sum of the number of zeros counted multiplicities of a non-zero element in the kernel of \(\CR j\), where interior zeros are counted twice. As the corresponding Maslov index is \(2\) on the disc component and \(4\) on the spheres, the kernel of \(\CR j\) is trivial due to the stability condition that the number of special points on each component \(C\) of \(S\) is at least \(3\). This argument uses elliptic regularity saying that the vector fields in \(\ker \CR j\) are smooth, see [54] . Therefore, one can show triviality of \(\ker \CR j=T_{\id }\GG _j\) (see Remark 6.4.1 ) alternatively using finiteness of \(\GG _j\) under the stability condition, see Remark 6.3.1 .

Using elliptic regularity as in [54] we see that the Cauchy–Riemann operator \(\CR j\co \Omega ^0\ra \Omega ^{0,1}_j\) is injective. The image is closed and has codimension \(2\big (\#A-\#D\big )\), cf. [43], Proposition 2.5 .

We provide \(\Omega ^{0,1}_j\) with the norm \(|y|_j\) given by the maximum of point-wise operator norms of \(y\) w.r.t. the metric \(g_j\). Because \(y\) is self-adjoint w.r.t. \(g_j\) the norm \(|y|_j\) is equal to the maximum of the square-root of the point-wise eigenvalues of the \(j\)-complex linear endomorphism field \(y\circ y\). In particular, the resolvent \((1-y)^{-1}\) of \(y\) at \(1\) is defined provided \(|y|_j<1\). We obtain a homeomorphism

\[ \mathfrak {i}\co \Omega ^{0,1}_j\cap B_1(0)\lra \JJ \,,\quad 0\longmapsto j \,, \]

defined on the open unit ball about zero via the conjugation

\[ y\longmapsto (1-y)j(1-y)^{-1} \,, \]

whose inverse is given by the

\[ k\longmapsto (k+j)^{-1}(k-j) \,, \]

cf. [7], Proposition 1.1.6 , [53], Proposition 2.6.4 or [61], p. 634 . Because \(\GG _j\) acts by isometries of \(g_j\) (see Remark 6.4.5 ) the conjugation map \(\mathfrak {i}\) is \(\GG _j\)-equivariant.

We claim that \(\JJ \) is a submanifold of the space \(\Omega ^1\) of all endomorphism fields of the tangent bundle of \(S\) and that \(\mathfrak {i}\) is a global chart. The case of almost complex structures compatible with \(\sigma \) essentially follows with the expositions in the above cited literature [7] , [53] , [61] . We will follow [31], Chapter I.7.3 taking the modifications for the case of almost complex structures only tamed by \(\sigma \) into account:

We call a not necessarily symmetric endomorphism field \(x\in \Omega ^1\) positive and write \(x>0\) provided that for all non-zero tangent vectors \(v\in TS\) the quadratic form \(g_j(v,xv)\) is positive. As the kernel of a positive endomorphism field \(x\in \Omega ^1\) is trivial we see that the inverse \(x^{-1}\in \Omega ^1\) exists. Therefore, for positive \(x\in \Omega ^1\) the endomorphism field \(1+x\) is positive as well such that the inverse \((1+x)^{-1}\) exists. In fact, the half space of all positive endomorphism fields \(\Omega ^1\cap \{x>0\}\) is an open cone in \(\Omega ^1\) closed under taking inverses. As above we provide \(\Omega ^1\) with the norm \(|x|_j\) given by the maximum of point-wise operator norms of \(x\in \Omega ^1\) w.r.t. the metric \(g_j\).

The Cayley transform

\[ \CC (x):=(1-x)(1+x)^{-1}=:\tilde {y} \]

defines a map

\[ \Omega ^1\cap \{x>0\}\lra \Omega ^1\cap B_1(0) \,. \]

Indeed, setting \(v=(1+x)^{-1}w\) polarisation yields \(4g_j(v,xv)=|w|_j^2-|\tilde {y}w|_j^2\), so that \(x\) is positive if and only if \(|\tilde {y}|_j<1\). We remark that there is an alternative formula

\[ \CC (x)=(1+x)^{-1}(1-x) \]

for the Cayley transform because \(1+x\) and \(1-x\) commute. Setting

\[ \CC (\tilde {y}):=(1-\tilde {y})(1+\tilde {y})^{-1} \]

we obtain a map

\[ \CC \co \Omega ^1\cap B_1(0)\lra \Omega ^1\cap \{x>0\} \]

in the converse direction. Again, this uses polarisation and that \(|\tilde {y}|_j<1\) implies triviality of the kernel of \(1+\tilde {y}\). Because the Cayley transform \(\CC \) is involutive the map \(\CC \co \Omega ^1\cap \{x>0\}\ra \Omega ^1\cap B_1(0)\) is a diffeomorphism.

Extending the conjugation map \(\mathfrak {i}\co \tilde {y}\mapsto (1-\tilde {y})j(1-\tilde {y})^{-1}\) to the open set \(\Omega ^1\cap B_1(0)\) yields a smooth map \(\Omega ^1\cap B_1(0)\ra \Omega ^1\). Observe, that \(\mathfrak {i}(\tilde {y})\) is a complex structure potentially reversing orientation. Restricting \(\mathfrak {i}\) to the \(j\)-complex anti-linear part \(\Omega ^{0,1}_j\) of \(\Omega ^1\) we get \(\mathfrak {i}(y)=j\CC (-y)\) for all \(y\in \Omega ^{0,1}_j\cap B_1(0)\). This defines an injection

\[ \mathfrak {i}=j\circ \CC \circ (-1)\co \Omega ^{0,1}_j\cap B_1(0)\lra \Omega ^1 \,. \]

In order to describe the image we observe that

\[ \CC \circ (-1)\co \Omega ^{0,1}_j\cap B_1(0)\lra \Omega ^1\cap \{x>0\}\cap \{jx=x^{-1}j\} \]

is a well defined homeomorphism with inverse \((-1)\circ \CC \). Additionally, the multiplication map \(j\co \Omega ^1\ra \Omega ^1\) is a diffeomorphism with inverse \(-j\) and restricts to the homeomorphism

\[ j\co \Omega ^1\cap \{x>0\}\cap \{jx=x^{-1}j\}\lra \JJ \,. \]

Indeed, positivity of \(x\) is equivalent to the positivity of \(\sigma _j(v,jxv)\) for all \(v\in TS\). Moreover, we find a smooth function \(f\) on the surface \(S\) such that the \(2\)-forms \(\sigma \) and \(\sigma _j\) satisfy \(\sigma =f\sigma _j\). Because \(\sigma (w,jw)=f|w|_j^2\) is positive for all non-zero \(w\in TS\) the function \(f\) must be positive. Hence, positivity of \(x\) is equivalent to the positivity of \(\sigma (v,jxv)\) for all \(v\in TS\), as \(\sigma _j\) and \(\sigma \) are positively proportional. It follows that the complex structure \(jx\) is tamed by \(\sigma \), i.e. preserves the orientation of the Riemann surface \((S,j)\). In total, the conjugation map

\[ \mathfrak {i}=j\circ \CC \circ (-1)\co \Omega ^{0,1}_j\cap B_1(0)\lra \JJ \]

is a well defined homeomorphism. The inverse is \(k\mapsto -\CC (-jk)=(k+j)^{-1}(k-j)\), where equality follows with the above alternative commuted formula for the Cayley transform. Furthermore \(\mathfrak {i}=j\circ \CC \circ (-1)\) is the restriction of the diffeomorphism obtained as the composite of \(\CC \circ (-1)\co \Omega ^1\cap B_1(0)\ra \Omega ^1\cap \{x>0\}\) with \(j\co \Omega ^1\ra \Omega ^1\) where defined. In other words, the inverse \((-1)\circ \CC \circ (-j)\) of \(j\circ \CC \circ (-1)\) serves as a global submanifold chart of \(\JJ \subset \Omega ^1\) the model being \(\Omega ^{0,1}_j\cap B_1(0)\subset \Omega ^1\).

In fact, \(\JJ \) is a complex manifold: A complex structure on the vector space \(\Omega ^{0,1}_j\) is given by \(y\mapsto jy\). An almost complex structure on \(\JJ \) is given by

\[\mathfrak {i}(y)=(1-y)j(1-y)^{-1}\]

on the tangent space \(T_{\mathfrak {i}(y)}\JJ =\Omega ^{0,1}_{\mathfrak {i}(y)}\) for all \(y\in \Omega ^{0,1}_j\cap B_1(0)\). Taking derivative of the equation

\[\mathfrak {i}(y)(1-y)=(1-y)j\]

w.r.t. \(y\) we get \(T_y\mathfrak {i}(\dot y)(1-y)-\mathfrak {i}(y)\dot y= -\dot y j \) and, using \(-\dot y j=j\dot y\), that

\[ T_y\mathfrak {i}(\dot y)= \big (j+\mathfrak {i}(y)\big )\dot y(1-y)^{-1} \]

for the linearisation of \(\mathfrak {i}\) at \(y\in B_1(0)\) for all tangent vectors \(\dot y\) of \(B_1(0)\subset \Omega ^{0,1}_j\). Using \(\mathfrak {i}(y)\big (j+\mathfrak {i}(y)\big )=\big (j+\mathfrak {i}(y)\big )j\) we obtain

\[ \mathfrak {i}(y)\circ T_y\mathfrak {i}=T_y\mathfrak {i}\circ j \,. \]

In other words, the linearisation \(T_y\mathfrak {i}\) is complex linear for all \(y\in \Omega ^{0,1}_j\cap B_1(0)\). Therefore, \(\mathfrak {i}\co \Omega ^{0,1}_j\cap B_1(0)\lra \JJ \) is biholomorphism.

We continue the considerations from Section 6.5 . For all \(j\in \JJ \) the operator \(j\CR j\co \Omega ^0\ra \Omega ^{0,1}_j\) is injective and

\[ H^1_j:=\Omega ^{0,1}_j/\im (j\CR j) \]

has real dimension \(2\big (\#A-\#D\big )\). Moreover, the operator \(j\CR j\) is \(\GG _j\)-equivariant by Remark 6.4.2 . Alternatively, argue with the theorema egregium and with Remark 6.4.5 as done in Remark 6.4.6 . Therefore, conjugation

\[ \psi ^*y:=T_{\psi }\psi ^{-1}\circ y_{\psi }\circ T\psi \]

by elements \(\psi \in \GG _j\) leaves \(\im (j\CR j)\) invariant, so that \(\GG _j\) induces an action on \(H^1_j\) via \(\psi ^*[y]:=[\psi ^*y]\).

Observe, that the complex structure \(y\mapsto jy\) on \(\Omega ^{0,1}_j\) introduced in Section 6.6 does not descent to a complex structure on the quotient \(H^1_j\). The reason is that the subspace \(\im (j\CR j)\) of \(\Omega ^{0,1}_j\) is not \(j\)-invariant as the operator \(j\CR j\co \Omega ^0\ra \Omega ^{0,1}_j\) is not complex linear, see Section 6.4 .

Choose a \(\GG _j\)-invariant complementary subspace \(E_j\subset \Omega ^{0,1}_j\) of \(\im (j\CR j)\) so that the quotient map \(y\mapsto [y]\) of \(\Omega ^{0,1}_j\) onto \(H^1_j\) restricts to a \(\GG _j\)-equivariant linear isomorphism \(E_j\ra H^1_j\).

Consider a so-called deformation of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\), which is a map

\[ \mathfrak {j}\co (V_j,0)\lra (\JJ ,j) \,,\qquad y\longmapsto j(y) \,, \]

defined on an open neighbourhood \(V_j\subset E_j\) of \(0\). If \(\mathfrak {j}\) is an embedding whose image \(\mathfrak {j}(V_j)\) is transverse to the orbits of \(\GG \), then the deformation \(\mathfrak {j}\) is called effective , cf. the [67], Definition 4.2.13 of a local slice through \(j\). Transversality can be expressed via the invertibility of the so-called Kodaira differential

\[ [T_y\mathfrak {j}]\co E_j\lra \Omega ^{0,1}_{j(y)}\lra H^1_{j(y)} \]

for all \(y\in V_j\), which is the composition of the linearisation \(T_y\mathfrak {j}\) with the quotient map \([\,.\,]\). Furthermore we call the deformation \(\mathfrak {j}\) symmetric if \(V_j\) is \(\GG _j\)-invariant (e.g. taking metric balls about \(0\in E_j\) w.r.t. to the \(L^{\infty }\)-norm \(|\,.\,|_j\) from the beginning of Section 6.6 or the \(L^2\)-norm \(\|\,.\,\|_j\) induced by \(\langle \,.\,,\,.\,\rangle _j\) from Example 6.7.1 ) and \(\mathfrak {j}\) is \(\GG _j\)-equivariant. The latter means that for all \(y\in V_j\) and \(\psi \in \GG _j\) we have that \(\psi ^*\big (\mathfrak {j}(y)\big )=\mathfrak {j}(\psi ^*y)\) so that

\[ \psi \co \big (S,j(\psi ^*y),D,\{1,\rmi ,-1\},A\big ) \lra \big (S,j(y),D,\{1,\rmi ,-1\},A\big ) \]

is a holomorphic isomorphism. Finally, the deformation \(\mathfrak {j}\) is called complex provided that the complementary subspace \(E_j\subset \Omega ^{0,1}_j\) is invariant under \(j\), i.e. is a complex linear subspace, and that \(\mathfrak {j}\) is holomorphic in the sense that the differential \(T\mathfrak {j}\) is complex linear, i.e. \(\mathfrak {j}(y)\circ T_y\mathfrak {j}=T_y\mathfrak {j}\circ j\) for all \(y\in V_j\subset E_j\).

In the situation of Section 6.7 we assume for the moment that the isotropy group \(\GG _j\) is trivial. Invertibility of the Kodaira differential \([T_y\mathfrak {j}]\) implies that the linear subspace \(T_y\mathfrak {j}(E_j)\) in \(\Omega ^{0,1}_{j(y)}\) is complementary to \(\im \big (j(y)\CR {j(y)}\big )\). With the arguments from [61], p. 634/5 and [67], Theorem 4.2.14 we obtain that

\[ \GG \times V_j\lra \JJ \,,\qquad (\varphi ,y)\longmapsto \varphi ^*\big (j(y)\big ) \,, \]

is a diffeomorphism onto a neighbourhood of the \(\GG \)-orbit of \(j=\id ^*\!\big (j(0)\big )\) for a sufficiently small open neighbourhood \(V_j\subset E_j\) of \(0\). In other words, the map \((\id ,y)\mapsto j(y)\) induces a homeomorphism

\[ (\id ,y)\longmapsto \big [S,j(y),D,\{1,\rmi ,-1\},A\big ]=\big [j(y)\big ] \]

defined on \(V_j\) onto a neighbourhood of \(\big [S,j,D,\{1,\rmi ,-1\},A\big ]=[j]\) in \(\RR _{\tau }=\JJ /\GG \). The inverse serves as a chart of a smooth manifold structure on \(\RR _{\tau }\).

For \(\GG _j\) non-trivial we give an equivariant version of the above construction: Assuming the situation of Section 6.7 we consider a symmetric effective deformation \(\mathfrak {j}\co (V_j,0)\ra (\JJ ,j)\), which we also denote by \(y\mapsto j(y)\). We would like to find a \(\GG \)-equivariant diffeomorphism

\[ \GG \times _{\GG _j} V_j\lra \JJ \,,\qquad (\varphi ,y)\longmapsto \varphi ^*\big (j(y)\big ) \,, \]

onto a neighbourhood of the \(\GG \)-orbit of \(j=\id ^*\!\big (j(0)\big )\), where \(\GG \times _{\GG _j} V_j\) is the quotient of \(\GG \times V_j\) by the action \((\varphi ,y)\mapsto \big (\psi ^{-1}\circ \varphi ,\psi ^*y\big )\), \(\psi \in \GG _j\).

In order to do so we would like to find an isomorphism

\[ \varphi \co \big (S,j(y),D,\{1,\rmi ,-1\},A\big ) \lra \big (S,k,D,\{1,\rmi ,-1\},A\big ) \]

for given \(k\in \JJ \) sufficiently close to \(j\in \JJ \) and for some \(y\in V_j\). Holomorphicity of \(\varphi \) translates to \(\varphi ^*k=j(y)\). In other words, \((\varphi ,y,k)\) is a zero of the non-linear Cauchy-Riemann operator

\[ F(\varphi ,y,k):=\frac 12\Big (T\varphi +k\circ T\varphi \circ j(y)\Big ) \,, \]

which is a section into the bundle \(\EE \ra \GG \times V_j\times \JJ \), whose fibre \(\EE _{(\varphi ,y,k)}\) is the vector space of sections of the bundle of complex anti-linear bundle homomorphisms from \(\big (TS,j(y)\big )\) to \((TS,k)\).

Setting \(F_k:=F(\,.\,,\,.\,,k)\) we will write the solutions \((\varphi ,y)\) of \(F_k(\varphi ,y)=0\) as a function of \(k\) via the implicit function theorem. The linearisation

\[ T_{(\id ,0)}F_j:\Omega ^0\times E_j\lra \Omega ^{0,1}_j \]

of \((\varphi ,y)\mapsto F_j(\varphi ,y)\) at \((\id ,0)\) equals

\[ (X,\dot y)\longmapsto \CR jX+\tfrac 12j\cdot \big (T_0\mathfrak {j}\big )(\dot y) \,. \]

With Sections 6.5 and 6.7 the operator

\[ -2j\cdot T_{(\id ,0)}F_j= -2j\CR j\oplus T_0\mathfrak {j} \]

is an isomorphism.

The implicit function theorem combined with an intermediate Sobolev completion and a subsequent elliptic regularity argument as in [61], p. 634/5 or in [67], Theorem 4.2.14 implies: There exists an open neighbourhood \(\KK \subset \JJ \) of \(j\), a possibly smaller \(\GG _j\)-invariant open neighbourhood \(V_j\subset E_j\) of \(0\), an open neighbourhood \(\HH \subset \GG \) of \(\id \) such that the sets \(\psi ^*\HH \) are pair-wise disjoint for all \(\psi \in \GG _j\), and a unique map

\[ \Phi \co \big (\KK ,j\big ) \lra \Big (\HH \times V_j,(\id ,0)\Big ) \]

such that

\[ F_k(\varphi ,y)=0 \quad \Longleftrightarrow \quad (\varphi ,y)=\Phi (k) \,, \]

whenever \((\varphi ,y,k)\in \HH \times V_j\times \KK \). Notice, that uniqueness implies \(\Phi \big (j(y)\big )=(\id ,y)\) for all \(y\in V_j\).

For all \(\psi \in \GG _j\) define a map

\[ \Phi _{\psi }\co \big (\KK ,j\big ) \lra \Big (\psi ^*\HH \times V_j, (\psi ,0)\Big ) \]

setting

\[ \Phi _{\psi }(k):= \psi ^*\big (\Phi (k)\big ) \,. \]

Observe that \(\Phi _{\psi }\big (j(y)\big )=\big (\psi ,\psi ^*y\big )\) for all \(y\in V_j\). Moreover, by symmetry of the deformation \(\mathfrak {j}\), which reads as \(\psi ^*\circ \mathfrak {j}=\mathfrak {j}\circ \psi ^*\) for all \(\psi \in \GG _j\), we get \(\psi ^*\circ F_k=F_k\circ \psi ^*\), and hence \(F_k\big (\Phi _{\psi }(k)\big )=0\) for all \(k\in \JJ \) and \(\psi \in \GG _j\). In other words, for all \(\psi \in \GG _j\) there exists a unique map \(\Phi _{\psi }\) with the above properties such that

\[ F_k(\varphi ,y)=0 \quad \Longleftrightarrow \quad (\varphi ,y)=\Phi _{\psi }(k) \,, \]

whenever \((\varphi ,y,k)\in \psi ^*\HH \times V_j\times \KK \).

We get the following global uniqueness statement : There exists potentially smaller neighbourhoods \(V_j\) and \(\KK \) such that for all solutions \((\varphi ,y,k)\in \GG \times V_j\times \KK \) of \(F(\varphi ,y,k)=0\) there exists a unique \(\psi \in \GG _j\) such that \(\varphi \in \psi ^*\HH \) and \((\varphi ,y)=\Phi _{\psi }(k)\). This follows arguing by contradiction with properness of the action \(\GG \times \JJ \ra \JJ \), \((\phi ,j)\mapsto \phi ^*j\), see Remark 6.3.2 .

Based on the current results of the implicit function theorem an orbifold chart about \(j\in \JJ \) can be obtained as follows: Let \(\UU _j\) be the image of \(\KK \) under the projection \([\,.\,]\co \JJ \ra \RR _{\tau }=\JJ /\GG \), i.e.

\[\UU _j=[\KK ]\,,\]

so that, in particular, \(\UU _j\) is open according to the quotient topology described in Remark 6.3.2 . With help of \(\Phi \) we find \(\varphi \in \HH \) and \(y\in V_j\) for each given \(k\in \KK \) such that \(\varphi ^*k=j(y)\). Hence, \([k]=\big [j(y)\big ]\), so that \(\UU _j\) is the set of all isomorphism classes \(\big [S,j(y),D,\{1,\rmi ,-1\},A\big ]=\big [j(y)\big ]\) with \(y\in V_j\).

The isotropy group \(\GG _j\) acts linearly on \(V_j\) by conjugation. In view of the metric obtained by restriction of the metric described in Example 6.7.1 this action is orthogonal. Hence, the action is effective , i.e. only for \(\id \in \GG _j\) all points of \(V_j\) are fixed points. The map

\[ \mathfrak {p}_j\co V_j/\GG _j\lra \UU _j \,,\qquad [y]\longmapsto \big [j(y)\big ] \]

is well-defined by symmetry of the deformation \(\mathfrak {j}\); \(\mathfrak {p}_j\) is continuous because the deformation \(\mathfrak {j}\) is and the respective quotient maps are open and continuous as explained in Remark 6.3.2 .

We claim that

\[ \UU _j\lra V_j/\GG _j \,,\qquad [k]\longmapsto \big [\Phi ^2(k)\big ] \,, \]

is the inverse map of \(\mathfrak {p}_j\), where \(\Phi ^2(k)\) denotes the second component of \(\Phi (k)\). First of all the map is well-defined by the following compatibility condition for uniformisers : Write \(\big [j(y_1)\big ]=[k]=\big [j(y_2)\big ]\) for \(y_1,y_2\in V_j\) and choose an isomorphism

\[ \phi \co \big (S,j(y_1),D,\{1,\rmi ,-1\},A\big ) \lra \big (S,j(y_2),D,\{1,\rmi ,-1\},A\big ) \,, \]

whose existence is guaranteed by the definition of the equivalence relation. We claim that

\[ \phi \in \GG _j \quad \text {and} \quad y_1=\phi ^*y_2 \,. \]

Indeed, we get \(F\big (\phi ,y_1,j(y_2)\big )=0\), so that we find a unique \(\psi \in \GG _j\) such that \(\phi \in \psi ^*\HH \) and \((\phi ,y_1)=\Phi _{\phi }\big (j(y_2)\big )\) by the above global uniqueness statement. Because of \(\Phi _{\phi }\big (j(y_2)\big )=(\psi ,\psi ^*y_2)\) we obtain \(\phi =\psi \in \GG _j\) and \(y_1=\psi ^*y_2\). Being well-defined follows now with the equation \(\Phi \big (j(y)\big )=(\id ,y)\) for \(y\in \{y_1,y_2\}\). Similarly, in order to verify the two-sided inverse property we obtain

\[ [y]\longmapsto \big [j(y)\big ]\longmapsto \Big [\Phi ^2\big (j(y)\big )\Big ]=[y] \]

and, writing \(\varphi ^*k=j(y)\) for \(y\in V_j\),

\[ [k]\longmapsto \big [\Phi ^2(k)\big ]=[y]\longmapsto \big [j(y)\big ]=[k] \,. \]

Therefore, the assignment \([k]\mapsto \big [\Phi ^2(k)\big ]\) is the inverse map \(\mathfrak {p}_j^{-1}\co \UU _j\lra V_j/\GG _j\). The inverse \(\mathfrak {p}_j^{-1}\) is continuous as well. This follows because \(k\mapsto \Phi ^2(k)\) is is continuous and the involved quotient maps are open and continuous, cf. Remark 6.3.2 . Consequently, the \(\GG _j\)-invariant map

\[ [\mathfrak {j}]\co V_j\lra \UU _j \]

induces a homeomorphism \(\mathfrak {p}_j\) form \(V_j/\GG _j\) onto \(\UU _j\). In other words, \(\big (V_j,\GG _j,\mathfrak {p}_j^{-1}\big )\) is an orbifold chart for \(\RR _{\tau }\) about \([j]=\big [S,j,D,\{1,\rmi ,-1\},A\big ]\) and \(\big (E_j,\GG _j,V_j,\UU _j,[\mathfrak {j}]\big )\) is a \(\tau \)- uniformiser by definition.

In order to describe the transformation behaviour of orbifold charts we consider symmetric effective deformations \(\mathfrak {j}\co V_j\ni y\mapsto j(y)\) and \(\mathfrak {k}\co V_k\ni z\mapsto k(z)\) of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) and \(\big (S,k,D,\{1,\rmi ,-1\},A\big )\), resp. About the respective \(\tau \)-uniformiser \(\big (E_j,\GG _j,V_j,\UU _j,[\mathfrak {j}]\big )\) and \(\big (E_k,\GG _k,V_k,\UU _k,[\mathfrak {k}]\big )\) we assume that \(\UU _j\cap \UU _k=\emptyset \). Hence, we find \(\varphi \in \GG \) such that \(\varphi ^*k=j\).

We claim that we can assume that \(j=j(0)=k(0)=k\). Indeed, define a deformation \(\mathfrak {k'}\co V_{k'}\ni z\mapsto k'(z)\),

\[ k'(z)=\varphi ^*\Big (k\big ((\varphi ^{-1})^*z\big )\Big ) \,, \]

of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\), whose domain is the subset \(V_{k'}:=\varphi ^*V_k\) of the complementary space \(E_{k'}:=\varphi ^*E_k\). For the latter use that \(g_{k'}\) and \(\varphi ^*g_k\) are conformally equivalent. By the naturality of the Kodaira differential we have

\[ [T_0\mathfrak {k'}]= \varphi ^*\circ [T_0\mathfrak {k}]\circ (\varphi ^{-1})^* \,, \]

so that \(\mathfrak {k'}\) is effective, see Remark 6.7.3 . In view of Remark 6.8.1 the deformation \(\mathfrak {k'}\) is symmetric because for all \(\psi \in \GG _k\), for which we have \(\psi ^*\circ \mathfrak {k}=\mathfrak {k}\circ \psi ^*\), it follows that \(\big (\varphi ^{-1}\circ \psi \circ \varphi \big )^*\circ \mathfrak {k'}= \mathfrak {k'}\circ \big (\varphi ^{-1}\circ \psi \circ \varphi \big )^*\).

Therefore, we consider deformations \(\mathfrak {j}\) and \(\mathfrak {k}\) such that \(j=j(0)=k(0)=k\). In view of the implicit function theorem above there exists a \(k(z)\)-holomorphic map \(\varphi (z)\in \GG \) close to \(\id \in \GG _j\) defined on \(\big (S,j\big (y(z)\big )\big )\) via the locally unique smooth map \(\Phi \big (k(z)\big )=\big (\varphi (z),y(z)\big )\) such that \(\Phi (j)=(\id ,0)\). Switching the roles results into a \(j(y)\)-holomorphic map \(\hat {\varphi }(y)\in \GG \) close to \(\id \in \GG _j\) defined on \(\big (S,k\big (z(y)\big )\big )\) via the locally unique smooth map \(\Phi \big (j(y)\big )=\big (\hat {\varphi }(y),z(y)\big )\) such that \(\Phi (j)=(\id ,0)\). Comparing both solutions using uniqueness yields

\[ \hat {\varphi }\big (y(z)\big )=\big (\varphi (z)\big )^{-1} \]

as well as \(z=z\big (y(z)\big )\) and \(y=y\big (z(y)\big )\). Therefore, after shrinking the domains according to the implicit function theorem if necessary, which results into \(\UU _j=\UU _k\), we obtain maps

\[ V_j\lra V_k \,,\quad y\longmapsto z(y) \qquad \text {and}\qquad V_k\lra V_j \,,\quad z\longmapsto y(z) \,, \]

which are smooth and inverse to each other such that \([\mathfrak {k}]\circ \big (y\mapsto z(y)\big )=\hat {\varphi }(y)^*[\mathfrak {j}]\). This results into a coordinate change of an orbifold structure because the construction is done in a \(\GG _j\)-equivariant fashion: For that use Remark 6.8.1 and observe that the solution set \(\big \{F_{k(z)}(\,.\,,y)=0\big \}\) is given by

\[ \Big \{\hat {\psi }\circ \varphi (z)\,\,\big |\,\,\hat {\psi }\in \GG _{k(z)}\Big \}= \big \{\psi ^*\varphi (z)\,\,|\,\,\psi \in \GG _{j,y}\big \} \,. \]

As for manifolds we obtain:

Referring to the current situation we define

\[ \mathbf {T}_{j,k}:= \Big \{ (\varphi ,y,z) \,\,\big |\,\, F_{k(z)}(\varphi ,y)=0 \Big \} \subset \GG \times V_j\times V_k \]

provided with the subspace topology and call the projection \(s\co \mathbf {T}_{j,k}\ra V_j\) onto \(V_j\) the source map ; the the projection \(t\co \mathbf {T}_{j,k}\ra V_k\) onto \(V_k\) the target map . These maps come with inverses \(y\mapsto \big (\hat {\varphi }^{-1}(y),y,z(y)\big )\) and \(z\mapsto \big (\varphi (z),y(z),z\big )\), resp., where \(\hat {\varphi }\big (y(z)\big )=\big (\varphi (z)\big )^{-1}\) as above. Hence, \(s\) and \(t\) are homeomorphisms providing \(\mathbf {T}_{j,k}\) with the structure of a smooth manifold of dimension \(2\big (\#A-\#D\big )\), and \(t\circ s^{-1}\) and \(s\circ t^{-1}\) correspond to the above transition maps \(y\mapsto z(y)\) and \(z\mapsto y(z)\), resp. By the properness argument in Remark 6.3.2 the map \(s\times t\co \mathbf {T}_{j,k}\ra V_j\times V_k\) is proper.

In other words, we obtain an étale proper Lie groupoid \((R_{\tau },\mathbf {R}_{\tau })\), which means the following: Take a sequence of \(\tau \)-uniformisers \(\big (E_{j_{\nu }},\GG _{j_{\nu }},V_{j_{\nu }},\UU _{j_{\nu }},[\mathfrak {j}_{\nu }]\big )\) such that \(\bigcup _{\nu }\UU _{j_{\nu }}\) covers \(\RR _{\tau }\). The objects are given by

\[ R_{\tau }:=\bigsqcup _{\nu }V_{j_{\nu }} \]

and the morphisms are

\[ \mathbf {R}_{\tau }:=\bigsqcup _{\nu ,\mu }\mathbf {T}_{j_{\nu },j_{\mu }}\,. \]

Morphisms can be composed whenever the corresponding target and source coincide. The resulting composition is smooth, has a unit and each morphism admits a smooth inverse. Furthermore all mentioned structure maps are smooth. The nodal Riemann moduli space \(\RR _{\tau }\) for the stable nodal type \(\tau \) appears now as orbit space

\[\RR _{\tau }=R_{\tau }/\!\sim \,,\]

where two objects between which there exists a morphism are considered to be equivalent.

A symmetric effective deformation \(\mathfrak {j}\) is called a skyscraper deformation if there exists a \(\GG _j\)-invariant neighbourhood \(U\subset S\) of \(\partial S\) together with the special points \(|D|\cup \{1,\rmi ,-1\}\cup A\) on which the deformation is stationary , i.e. if \(j(y)=j\) restricted to \(U\) for all \(y\in V_j\). In view of the examples of symmetric effective deformations in Section 6.6 and Remark 6.6.1 skyscraper deformations can be obtained by restriction of symmetric effective deformations to a complementary \(\GG _j\)-invariant vector space \(E_j\) of \(\im (j\CR j)\) whose elements vanish on \(U\).

In order to construct such a vector space \(E_j\) we denote by \(\XX \) the space of all smooth vector fields on \(S\) that are tangent to the boundary along \(\partial S\) and admit \(3\) zeros on each connected component of \(S\) equal to special points in \(|D|\cup \{1,\rmi ,-1\}\cup A\); on the disc component the zeros are required to be \(1,\rmi ,-1\). The operator \(j\CR j\) restricted to \(\XX \) induces an isomorphism \(\XX \ra \Omega ^{0,1}_j\).

We begin with the un-isotropic situation \(\GG _j=\{\id \}\). In order to construct a complement of \(\Omega ^0\) in \(\XX \) we write \(z_k\) for the elements of \(|D|\cup A\) and choose local holomorphic charts \((\C ,\rmi )\) for \((S,j)\) about the special points \(z_k\). We require that the chart domains are mutually disjoint and contained in \(S\setminus \partial S\). Let \(f_k\) be smooth cut off functions on \(S\) that have their supports in the interior of \(r\)-disc neighbourhoods about \(z_k\) w.r.t. \(g_j\) contained in the chosen chart domains; the \(f_k\) are required to by constantly \(1\) on the \(r/2\)-disc neighbourhoods about \(z_k\). Given \(X\in \XX \) we define vector fields \(X_k\) on \(S\). We require that the \(X_k\) are given by \(X(z_k)f_k\) in the chosen charts extended by zero to \(S\). Observe that the \(X_k\) vanish for special points that correspond to the zeros defining \(\XX \). Moreover, the \(X_k\) are holomorphic on the \(r/2\)-discs.

Let \(P\co \XX \ra \XX \) be the projector, i.e. \(P^2=P\), given by

\[ P(X):=X-\sum _kX_k \,. \]

Observe that \(P\) restricts to the identity on \(P(\XX )=\Omega ^0\). The desired complement of \(\Omega ^0\) in \(\XX \) is \((1-P)(\XX )\) as \(1-P\) is a projector as well. The elements of \(j\CR j(1-P)(\XX )\) vanish on a neighbourhood of all special points \(|D|\cup \{1,\rmi ,-1\}\cup A\) and of \(\partial S\). Moreover, the dimension of \((1-P)(\XX )\) equals

\[ 2\big (\#A-\#D\big ) \]

by the result of the computation of Section 6.5 multiplied by \(-1\).

Finally we set \(E_j:=j\CR j(1-P)(\XX )\). The elements of \(E_j\) vanish on a neighbourhood of the union of the special points \(|D|\cup \{1,\rmi ,-1\}\cup A\) and of the boundary \(\partial S\). Furthermore the isomorphism \(j\CR j\co \XX \ra \Omega ^{0,1}_j\) sends the splitting \((1-P)(\XX )\oplus \Omega ^0\) of \(\XX \) to the splitting \(E_j\oplus \im (j\CR j)\) of \(\Omega ^{0,1}_j\).

We treat the case of a non-vanishing isotropy group \(\GG _j\), which acts by permutations on \(\{z_k\}=|D|\cup A\). It suffices to change the above projector \(P\) by replacing the vector fields \(X_k\) by \(\GG _j\)-invariant vector fields \(\hat {X}_k\). For that denote by \(B_r(z_k)\), \(r>0\), the interior of the above \(r\)-discs. Observe that the disjoint union of the \(B_r(z_k)\) is \(\GG _j\)-invariant as \(\GG _j\) acts by isometries on \((S,g_j)\).

For each \(z_k\) we assign a partner point \(w_k\in B_{r/2}(z_k)\setminus \{z_k\}\) requiring that the \(\psi (w_k)\) are pair-wise distinct for all \(\psi \in \GG _j\) and all \(k\). Notice, that the distance between \(z_k\) and its partner \(w_k\) is \(\GG _j\)-invariant for all \(k\). We choose \(\varepsilon >0\) such that \(B_{\varepsilon }(w_k)\subset B_{r/2}(z_k)\setminus \{z_k\}\) for all \(k\). Furthermore, we reqiure that the \(\psi \big (B_{\varepsilon }(w_k)\big )\) are pair-wise disjoint for all \(\psi \in \GG _j\) and for all \(k\). Modify the cut off functions \(f_k\) so that \(f_k\) has support in

\[ B_r(z_k)\,\,\setminus \quad \bigsqcup _{\ell \neq k\,\,\text {and}\,\,\psi \in \GG _j} \psi \big (\overline {B_{\varepsilon /2}(w_{\ell })}\big ) \]

and is equal to \(1\) on

\[ B_{r/2}(z_k)\,\,\setminus \quad \bigsqcup _{\ell \neq k\,\,\text {and}\,\,\psi \in \GG _j} \psi \big (\overline {B_{\varepsilon }(w_{\ell })}\big ) \,. \]

In particular, for all \(k\), we get \(f_k(w_k)=1\) and \(f_k\big (\psi (w_{\ell })\big )=0\) for all \(\ell \neq k\) and \(\psi \in \GG _j\). With the cut off functions \(f_k\) modified we define \(X_k\) for given \(X\in \XX \) as in the un-isotropic case.

We define the symmetrisations via

\[ \hat {X}_k:=\sum _{\psi \in \GG _j}\psi ^*X_k \,. \]

We have \(\phi ^*\hat {X}_k=\hat {X}_k\) for all \(\phi \in \GG _j\) and for all \(k\) because \(\GG _j\) acts on itself via composition permuting \(\GG _j\). The \(\hat {X}_k\) that are assigned to the zeros \(z_k\) of \(\XX \) vanish; the remaining \(\hat {X}_k\) span a \(2\big (\#A-\#D\big )\)-dimensional vector space because

\[ \hat {X}_k(w_{\ell })=X_{\ell }(w_{\ell }) \]

for all \(k,\ell \). A basis can be obtained by taking \(X\in \XX \) with \(X(z_k)\) non-zero, so that the corresponding partners \(X_k(w_k)\) do not vanish.

Consequently, we obtain a complex version of Proposition 6.8.2 , so that, in particular, \(\RR _{\tau }\) is orientable.

For a complex number \(a\) of modulus \(|a|\leq 1\) we consider the intersection of the planar algebraic curve \(\{zw=a\}\subset \C \times \C \) with the polydisc \(\D \times \D \). For \(a=0\) this curve is the union of the discs \(\{z=0\}=\{0\}\times \D \) and \(\{w=0\}=\D \times \{0\}\) that intersect in the singularity of the curve. For \(a\neq 0\) the equation \(zw=a\) can be solved by \(w=a/z\) so that we obtain a cylinder that has no singularities: The restriction of the projection \((z,w)\mapsto z\) to the curve \(\{zw=a\}\) yields a biholomorphism onto the annulus \(\{|a|\leq |z|\leq 1\}\) in the first coordinate plane. Interchanging \(z\) and \(w\) yields a biholomorphism onto \(\{|a|\leq |w|\leq 1\}\). Both biholomorphisms constitute holomorphic charts of \(\{zw=a\}\). The transition map from the first annulus to the second is

\[z\longmapsto \frac {a}{z}\,.\]

Taking positive and negative holomorphic polar coordinates \((z,w)\mapsto -\ln z\) and \((z,w)\mapsto \ln w\), resp., i.e. writing

\[ z=\rme ^{-(s+\rmi t)} \qquad \text {and}\qquad w=\rme ^{u+\rmi v} \,, \]

the transition map gets

\[ \big [0,-\ln |a|\big ]\times S^1\lra \big [\ln |a|,0\big ]\times S^1 \,,\qquad (s,t)\longmapsto \big (s+\ln |a|,t+\arg a\big ) \,, \]

where \(S^1=\partial \D \). For the complex logarithm we use the main branch.

Observe that rotations \(z\mapsto \rme ^{-\rmi \theta _+}z\) and \(w\mapsto \rme ^{\rmi \theta _-}w\) for \(\theta _+,\theta _-\in S^1\) of the coordinate planes, which correspond to

\[ (s,t)\longmapsto \big (s,t+\theta _+\big ) \qquad \text {and}\qquad (s,t)\longmapsto \big (s,t+\theta _-\big ) \]

w.r.t. positive and negative holomorphic polar coordinates, resp., result into a change of the defining equation to

\[ zw=\rme ^{-\rmi (\theta _+-\theta _-)}a \]

as the pull back along \((z,w)\mapsto \big (\rme ^{-\rmi \theta _+}z,\rme ^{\rmi \theta _-}w\big )\) yields \(zw=a\). The coresponding transition map is

\[ (s,t)\longmapsto \Big (s+\ln |a|,t+\arg a-\big (\theta _+-\theta _-\big )\Big ) \,. \]

A switch of the coordinates \((z,w)\mapsto (w,z)\) does not effect the proceeding consideration.

Given \(\big [S,j,D,\{1,\rmi ,-1\},A\big ]\in \RR _{\tau }\) we describe a similar desingularisation about a nodal pair \(\{z_0,w_0\}\in D\) in terms of parametrised connected sum . Choose a small disc structure \(\mathbf {D}_j\) on \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\). Denote the corresponding discs about the nodal points \(z_0,w_0\in |D|\) by \(D_{z_0}\) and \(D_{w_0}\), resp., and choose boundary points \(z_{\partial }\in \partial D_{z_0}\) and \(w_{\partial }\in \partial D_{w_0}\). We call the pair \(\{z_{\partial },w_{\partial }\}\) a decoration of the nodal pair \(\{z_0,w_0\}\). By [54], Theorem C.5.1 there exists unique biholomorphic identifications of \(\big ((D_{z_0},z_0,z_{\partial }),j\big )\) and \(\big ((D_{w_0},w_0,w_{\partial }),j\big )\), resp., with \(\big ((\D ,0,1),\rmi \big )\).

For given gluing parameter \(a\in \D \), \(a\neq 0\), replace \(-\ln |a|\) by the modulus

\[ R=\rme ^{1/|a|}-\rme \]

in the discussion about the planar algebraic curve \(\{zw=a\}\). Identify the first annulus \(\{\rme ^{-R}\leq |z|\leq 1\}\) with the second \(\{\rme ^{-R}\leq |w|\leq 1\}\) via the transition map

\[ z\longmapsto \frac {\rme ^{-R+\rmi \arg (a)}}{z} \,, \]

which w.r.t. positive and negative holomorphic polar coordinates reads as

\[ [0,R]\times S^1\lra [-R,0]\times S^1 \,,\qquad (s,t)\longmapsto \big (s-R,t+\arg a\big ) \,. \]

We obtain a surface \(S_a\) from \(S\setminus \big (\Int (D_{z_0})\cup \Int (D_{w_0})\big )\) by gluing the finite cylinder \(Z_a:=[0,R]\times S^1\), which is identified with \([-R,0]\times S^1\) via the above transition map, along the respective boundary circles via the restrictions of the biholomorphic identifications of \(D_{z_0}\) and \(D_{w_0}\), resp, with \(\D \).

The construction of the surface \(S_a\) defines a complex structure \(j_a\) that coincides with \(j\) on \(S\setminus \big (\Int (D_{z_0})\cup \Int (D_{w_0})\big )\) and with \(\rmi \) on the cylinder \(Z_a\) of modulus \(R\). This results into an element \(\big [S_a,j_a,D_a,\{1,\rmi ,-1\},A\big ]\) of \(\RR _{\tau '}\) with stable nodal type \(\tau '\), which necessarily differs from \(\tau \). The respective special points are given by

\[ D_a:=D\setminus \big \{\{z_0,w_0\}\big \} \,, \]

\(\{1,\rmi ,-1\}\), and \(A\) under the inclusion of \(S\setminus \big (\Int (D_{z_0})\cup \Int (D_{w_0})\big )\) into \(S_a\).

A change of biholomorphic identifications of \(D_{z_0}\) and \(D_{w_0}\) with \(\D \) is given by a rotation of the boundary points \(z_{\partial }\) and \(w_{\partial }\), resp., which in coordinates reads as \(z\mapsto \rme ^{-\rmi \theta _+}z\) and \(w\mapsto \rme ^{\rmi \theta _-}w\), say. Gluing with the rotated identifications yields a biholomorphic copy \(\big (S_b,j_b,D_b,\{1,\rmi ,-1\},A\big )\) of \(\big (S_a,j_a,D_a,\{1,\rmi ,-1\},A\big )\), where

\[ b=\rme ^{-\rmi (\theta _+-\theta _-)}a \,. \]

To obtain a biholomorphic map take the identity map on \(S\setminus \big (\Int (D_{z_0})\cup \Int (D_{w_0})\big )\) and the rotated transition map \(Z_a\ra Z_b\) given by

\[ (s,t)\longmapsto \Big (s-R,t+\arg a-\big (\theta _+-\theta _-\big )\Big ) \]

on \(Z_a\).

Such rotations naturally appear when a automorphism \(\psi \in \GG _j\) for which the nodal points \(z_0\) and \(w_0\) are fixed-points, i.e. \(\psi (z_0)=z_0\) and \(\psi (w_0)=w_0\), acts on \(S\). Indeed, \(\psi \) preserves the complement of \(D_{z_0}\cup D_{w_0}\) in \(S\) and induces rotations on \(D_{z_0}\cup D_{w_0}\). The rotations are measured by the change of decorations from \(\{z_{\partial },w_{\partial }\}\) to \(\psi \big (\{z_{\partial },w_{\partial }\}\big )\) in terms of angles \(-\theta _+\) and \(\theta _-\), say. Therefore, we get a holomorphic diffeomorphism

\[ \big (S_a,j_a,D_a,\{1,\rmi ,-1\},A\big ) \lra \big (S_b,j_b,D_b,\{1,\rmi ,-1\},A\big ) \]

as above which this time coincides with \(\psi \) on \(S\setminus \big (\Int (D_{z_0})\cup \Int (D_{w_0})\big )\).

We denote by

\[ \D ^D \]

the set of all maps from the set of nodal points \(D\) to the set \(\D \) of complex numbers of modulus less than or equal to \(1\). Choose a small disc structure \(\mathbf {D}_j\) on \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) together with a decoration for each disc in \(\mathbf {D}_j\). The choice of decorations determine holomorphic diffeomorphisms of all discs of the disc structure with \(\D \) such that the nodal point is mapped to \(0\in \D \) and the decoration to \(1\in \D \). Given \(\bfa \in \D ^D\) we perform the described parametrised connected sum about each nodal pair \(\{z,w\}\in D\) with gluing parameter

\[ a_{\{z,w\}}:=\bfa \big (\{z,w\}\big ) \,. \]

This is done by replacing the node \(\{z,w\}\) with the cylinder \(Z_{a_{\{z,w\}}}^{\{z,w\}}\). In the case of a vanishing gluing parameter \(a_{\{z,w\}}\) formally

\[ Z_0^{\{z,w\}}:=D_z\sqcup D_w \]

is given by the disjoint union of half-infinite cylinders \([0,\infty )\times S^1\) and \((-\infty ,0]\times S^1\) of infinite modulus after removing the nodal points \(z\) and \(w\). In other words, if \(a_{\{z,w\}}=0\) we do nothing and keep the nodal pair \(\{z,w\}\in D_{\bfa }\), so that \(D_{\bfa }\) arises from \(D\) by removing all nodal pairs \(\{z,w\}\) with \(a_{\{z,w\}}\neq 0\). The resulting surface is denoted by

\[ \big (S_{\bfa },j_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ) \,. \]

Starting off with a skyscraper deformation \(V_j\ni y\mapsto j(y)\) and a small disc structure of sufficiently small discs we will get

\[ \big (S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ) \]

by the same construction.

In order to describe the effect of the \(\GG _j\)-action of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) on the desingularisation we denote by \(\kappa _{z,w}\), \(\{z,w\}\in D\), the complex anti-linear map \(T_wS\ra T_zS\) that conjugated with the linearisations of the biholomorphic identifications of the discs \(D_w\) and \(D_z\) with \(\D \) is equal to the complex conjugation map \(x+\rmi y\mapsto x-\rmi y\) on \(\C \). Interchanging the role of \(z\) and \(w\) replaces \(\kappa _{z,w}\) by its inverse \(\kappa _{w,z}=(\kappa _{z,w})^{-1}\). We call \(\kappa _{z,w}\) a compatible nodal identifier . Given \(\psi \in \GG _j\) and \(\{z,w\}\in D\) we define the phase function

\[ \Theta _{\{z,w\}}(\psi )\co T_zS\lra T_zS \]

by

\[ \Theta _{\{z,w\}}(\psi ):= \kappa _{z,w}\circ T_{\psi (w)}(\psi )^{-1}\circ \kappa _{\psi (w),\psi (z)}\circ T_z\psi \,. \]

Taking positive and negative holomorphic polar coordinates about \(z\) and \(w\), resp., so that \(\psi \) acts in coordinates by multiplication with \(\rme ^{-\rmi \theta _+}\) and \(\rme ^{\rmi \theta _-}\), resp., we get

\[ \big (\Theta _{\{z,w\}}(\psi )\big )(v)= \rme ^{-\rmi (\theta _+-\theta _-)}v \]

for all \(v\in T_zS\), which we simply declare to a multiplication operator

\[ \Theta _{\{z,w\}}(\psi )\equiv \rme ^{-\rmi (\theta _+-\theta _-)} \,. \]

This shows independence of the phase function

\[ \Theta \co D\times \GG _j\lra S^1 \,,\qquad \big (\{z,w\},\psi \big )\longmapsto \Theta _{\{z,w\}}(\psi ) \,, \]

of the chosen ordering of \(\{z,w\}\) in the definition of \(\Theta _{\{z,w\}}(\psi )\) and of the chosen parity of the holomorphic polar coordinates about \(\psi (z)\) and \(\psi (w)\). This results in a \(\GG _j\)-action on \(\D ^D\) defined by \(\psi _*\bfa =\bfb \) via

\[ b_{\{\psi (z),\psi (w)\}}:= \Theta _{\{z,w\}}(\psi )\cdot a_{\{z,w\}} \]

for all \(\{z,w\}\in D\). Consequently, for any skyscraper deformation \(V_j\ni y\mapsto j(y)\), a small disc structure of sufficiently small discs, and \(\psi \in \GG _j\) we get an isomorphism

\[ \psi _{\bfa }\co \Big (S_{\bfa },j(\psi ^*y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\Big ) \lra \Big (S_{\psi _*\bfa },j(y)_{\psi _*\bfa },D_{\psi _*\bfa },\{1,\rmi ,-1\},A\Big ) \]

by the gluing construction and symmetry of \(y\mapsto j(y)\).

A neighbourhood base of a second countable paracompact Hausdorff topology on \(\RR _N\), \(N\geq 0\), is given by the family of subsets of \(\RR _N\), whose elements are of the form

\[ \big [S_{\bfa },k_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ] \]

with \(N=\#A\), which are obtained from a nodal disc \(\big (S,k,D,\{1,\rmi ,-1\},A\big )\) by the parametrised connected sum construction with given decorated small disc structure \(\mathbf {D}_j\), with gluing parameter \(\bfa \in \D ^D\) with \(|\bfa |<\varepsilon \) for some \(\varepsilon \in (0,1)\), with complex structures \(k\) that belong to an open neighbourhood of \(j\) in \(\JJ \) such that \(k=j\) restricted to \(\mathbf {D}_j\). This follows as in [43], Proposition 2.4 and [42], Theorem 2.15 and Theorem 5.13 because no extra argument for boundary un-noded nodal discs is needed caused by absence of boundary nodes. The Hausdorff property follows with Gromov compactness for stable holomorphic discs, see [25] .

The induced topology on \(\RR _{\tau }\) in \(\RR _N\) agrees with the one on \(\RR _{\tau }\) previously defined in Remark 6.3.2 . The induced notion of convergence of sequences in \(\RR _N\) coincides with Gromov convergence as described in [1], Chapter 1 , [66], Appendix B or in [15], Section 4 , [46], Chapter IV after Schwartz reflection along the boundary of the nodal discs for example.

In order to obtain an orbifold structure on \(\RR _N\) we consider desingularisations

\[ \big (S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ) \]

of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) as described in Section 6.10 . For \(\bfa _0\in \D ^D\) consider the set \(D\setminus D_{\bfa _0}\) of all nodal pairs \(\{z,w\}\in D\) on which the map \(\bfa _0\) is non-zero. Define a deformation

\[ \mathfrak {j}_{\bfa _0}\co V_{j_{\bfa _0}}\times \D ^{D\setminus D_{\bfa _0}} \lra \JJ _{S_{\bfa _0}} \,,\qquad (y,\bfb )\longmapsto j(y)_{\bfa _0+\bfb } \,, \]

of

\[ \big (S_{\bfa _0},j(y)_{\bfa _0},D_{\bfa _0},\{1,\rmi ,-1\},A\big ) \]

by setting \(\bfa =\bfa _0+\bfb \). For small deformation parameter \(\bfb \) the deformed family of surfaces equals

\[ \Big (S_{\bfa _0+\bfb },j(y)_{\bfa _0+\bfb },D_{\bfa _0+\bfb },\{1,\rmi ,-1\},A\Big ) \,. \]

The nodal discs family is isomorphic to

\[ \Big (S_{\bfa _0},j'(y)_{\bfb },D_{\bfa _0},\{1,\rmi ,-1\},A\Big ) \]

with corresponding deformation

\[ \mathfrak {j}'_{\bfb }\co V_{j_{\bfa _0}}\times \D ^{D\setminus D_{\bfa _0}} \lra \JJ _{S_{\bfa _0}} \,,\qquad (y,\bfb )\longmapsto j'(y)_{\bfb } \,, \]

via an isomorphism that is the identification map on the complement of the respective small disc structure, so that the deformation is given by rotations and stretchings of the cylindrical neck regions that correspond to the nodes, on which \(\bfa _0\) not vanishes. The partial Kodaira differential of \(\mathfrak {j}'_{\bfb }\) at \((y,0)\) is

\[ \big [T_{(y,0)}\mathfrak {j}'_{\bfb }\big ]\co E_{j_{\bfa _0}}\times \C ^{D\setminus D_{\bfa _0}} \lra \Omega ^{0,1}_{{j(y)}_{\bfa _0}}\lra H^1_{{j(y)}_{\bfa _0}} \,. \]

Similarly to [43], Theorem 2.13 one constructs uniformisers of an orbifold structure on \(\RR _N\) as the above desingularisations stay away from the boundary of the nodal discs in \(\RR _N\). For given \(\big [S,j,D,\{1,\rmi ,-1\},A\big ]\in \RR _N\) such a uniformiser is a deformation

\[ \VV \ni (y,\bfa )\longmapsto \big (S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ) \]

of \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) for an open subset \(\VV \) of \(V_j\times \D ^D\) such that the following holds:

• The union of all equivalence classes \(\big [S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ]\) over all \((y,\bfa )\in \VV \) is an open subset of \(\RR _N\).
• The map \(\VV \ra \UU \) that assigns to \((y,\bfa )\) the class \(\big [S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\big ]\) descends to a homeomorphism \(\VV /\GG _j\ra \UU \).
• An isomorphism between the classes belonging to \((y,\bfa ),(z,\bfb )\in \VV \) is given by \(\psi _{\bfa }\) for \(\psi \in \GG _j\) and \((z,\bfb )=\big (\psi _*y,\psi _*\bfa \big )\).
• For all points in \(\VV \) the partial Kodaira differential is an isomorphism.

Compatibility of uniformisers is expressed via the sets

\[ \mathbf {T}_{j,k}:= \Big \{ \big (\varphi ,(y,\bfa ),(z,\bfb )\big ) \Big \} \subset \GG \times (V_j \times \D ^D)\times (V_k \times \D ^D) \]

corresponding to all isomorphisms

\[ \varphi \co \Big (S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},A\Big ) \lra \Big (S_{\bfb },k(z)_{\bfb },D_{\bfb },\{1,\rmi ,-1\},A\Big ) \,, \]

which are smooth manifolds of dimension \(2\#A\), so that \(\RR _N\) supports an étale proper Lie groupoid structure as formulated after Proposition 6.8.2 . This follows with the (anti-)gluing construction ( [43], Section 2.4 ) for the non-linear Cauchy–Riemann operator along the nodes (which take place away from the boundary) known from Floer theory, cf. [43], Theorem 2.16 and [42], Theorem 2.24 . Similarly, the universal property of the construction stated in [43], Theorem 2.16 translates into the present situation. The involved variation of marked points can be treaded as in [42], Remark 3.17 . Finally, using convex interpolation between the exponential gluing profile \(\rme ^{1/r}-\rme \) we used in the gluing construction and the logarithmic gluing profile \(-\ln r\) that appeared in the desingularisation of the complex algebraic curve at the beginning of Section 6.10 naturally yields an orientation on \(\RR _N\) that extends the complex orientation on \(\RR _{\tau }\) given in Proposition 6.9.4 , see [42], Section 2.3.2 .

We consider the tame almost complex manifold \(\big (\hat {W},\hat {\Omega },\hat {J}\big )\) defined in Section 5.2 . For boundary un-noded nodal discs \((S,j,D)\) as introduced in Section 6.1 we consider smooth maps

\[ u\co (S,\partial S)\lra (\hat {W},N^*) \]

that descend to continuous maps on \(S/D\). If \(D\) is empty we call \(u\) un-noded . If in addition \(Tu\circ j=\hat {J}(u)\circ Tu\) we call \(u\) a nodal holomorphic disc map . Observe that we do not need to consider nodal points on the boundary due to the Gromov compactification described in Remark 4.4.1 .

More generally, we consider continuous maps \(u\co (S,\partial S)\ra (\hat {W},N^*)\) defined on a marked boundary un-noded nodal disc \(\big (S,j,D,\{m_0,m_1,m_2\}\big )\) (see Section 6.1 ) such that \(u\) descends to a continuous map on the quotient \(S/D\) and such that \(u\big (\!\Int S\big )\subset \Int \hat {W} \). Moreover, we require that \(u\) is contained in the Sobolev space of square integrable maps

\[ H^{3,\sigma }(S,j)\equiv H^{3,\sigma }\big (S,j,D,\{m_0,m_1,m_2\}\big ) \]

following [43], Definition 1.1 : We require that \(u\) is of class \(u\in H_{\loc }^3\big (S\setminus |D|\big )\) and that w.r.t. positive holomorphic polar coordinates \([0,\infty )\times S^1\), \(S^1=\R /2\pi \Z \), about the nodal points \(|D|\) (see Section 6.10 ) the map \(u\) is of weighted Sobolev class \(H^{3,\sigma }\). The weights are given by \(\rme ^{\sigma s}\), \(s\in [0,\infty )\), for some \(\sigma \in (0,1)\). In other words, \(u\) is contained in \(H^{3,\sigma }\) precisely if all weak derivatives \(D^{\alpha }u\), \(|\alpha |\leq 3\), of \(u\) on \([0,\infty )\times S^1\) exist and all \(D^{\alpha }u\cdot \rme ^{\sigma s}\), \(|\alpha |\leq 3\), are square integrable on \([0,\infty )\times S^1\). The latter is equivalent to \(u\,\rme ^{\sigma s}\in H^3\) on \([0,\infty )\times S^1\). In particular, by Sobolev embedding, \(u\) is \(C^1\) (up to the boundary \(\partial S\)) restricted to \(S\setminus |D|\). But in general \(u\) is not differentiable at the nodal points \(|D|\) on \(S\). Consider for example the continuous function \(u(z)=|z|^{\frac {1+\sigma }{2}}\) in holomorphic coordinates \(z\in \C \), which w.r.t. positive holomorphic polar coordinates reads as \((s,t)\mapsto \rme ^{-\frac {1+\sigma }{2}s}\).

The space \(H^{3,\sigma }(S,j)\) is well defined, i.e. invariant under coordinate changes after possibly shrinking the chart domains. Away from the nodes \(|D|\) this follows as for \(H_{\loc }^3\big (S\setminus |D|\big )\) via [4], Theorem 3.41 . Near the nodes we observe that the area form \(\rme ^{2\sigma s}\rmd t\wedge \rmd s\) w.r.t. positive holomorphic polar coordinates corresponds to the singular area form \(|z|^{-2(1+\sigma )}\frac {\rmi }{2}\rmd z\wedge \rmd \bar {z}\) in holomorphic coordinates about the nodal point \(0\in \C \). The area form \(\frac {\rmi }{2}\rmd z\wedge \rmd \bar {z}\) transforms under biholomorphic coordinate changes via a conformal factor, which we can assume to be bounded above and away from zero by shrinking the chart domains if necessary. The coordinate change itself is of the form \(z\mapsto zh(z)\), where \(0\) corresponds to a nodal point. Here \(h\) is a holomorphic function, whose absolute value can be assumed to be bounded above and away from zero also. Consequently, the singular area form \(|z|^{-2(1+\sigma )}\frac {\rmi }{2}\rmd z\wedge \rmd \bar {z}\) transforms via a bounded above and away from zero conformal factor also. Hence, the same holds true for \(\rme ^{2\sigma s}\rmd t\wedge \rmd s\). In fact, the above coordinate change becomes

\[ (s,t)\longmapsto (s,t)-\ln \Big (h\big (\rme ^{-(s+\rmi t)}\big )\Big ) \,, \]

whose derivatives are bounded above and whose first derivative is bounded away from zero. Therefore, invariance under coordinate changes near the nodes follows as in [4], Theorem 3.41 . By the same arguments we see that locally defined norms on \(H_{\loc }^3\) and \(H^{3,\sigma }\) transform via the respective coordinate changes to equivalent norms. This defines a topology on \(H^{3,\sigma }(S,j)\); a neighbourhood base is given by the set of those maps that restricted to one of the above charts belong to an open set in \(H_{\loc }^3\) and \(H^{3,\sigma }\), resp.

To each \(u\in H^{3,\sigma }(S,j)\) we assign the symplectic energy integral

\[ \int _Su^*\hat {\Omega } \]

by approximating the continuous map \(u\) by a \(C^1\)-map \(v\) and defining the symplectic energy integral via \(\int _Su^*\hat {\Omega }:=\int _Sv^*\hat {\Omega }\). This is well defined and, in fact, by Stokes theorem, independent of the choice of representative of the homology class \([u]\) in \(\hat {W}\) relative \(u(\partial S)\subset N^*\). This can be seen as follows: Taking approximations \(v\) of \(u\) that are equal to \(u\) restricted to the complement of disc like neighbourhoods \(B_r(0)\) in \(S\) of the nodal points \(0\in |D|\) the symplectic energy integral is given by \(\int _{S\setminus |D|}u^*\hat {\Omega }\). Indeed, take \(r>0\) so small such that the \(B_r(0)\) are contained in pair-wise disjoint chart domains of \(S\) about the nodal points \(0\) in \(|D|\) and such that the \(u\big (B_r(0)\big )\) are contained in pair-wise disjoint ball like chart domains of \(\hat {W}\). By Stokes theorem decomposing \(B_r(0)=\big (B_r(0)\setminus B_{\varepsilon }(0)\big )\cup B_{\varepsilon }(0)\) it suffices to show that the integrals

\[ \int _{B_{\varepsilon }(0)}u^*\hat {\Omega } \qquad \text {and} \qquad \int _{\partial B_{\varepsilon }(0)}u^*\lambda \]

converge to zero as \(\varepsilon \in (0,r)\) tends to \(0\), where \(\lambda \) is a local primitive of \(\hat {\Omega }\) defined on the ball like neighbourhoods of \(u(|D|)\) in \(\hat {W}\). By the transformation formula we can compute the integrals w.r.t. positive holomorphic polar coordinates via

\[ \int _{(R,\infty )\times S^1}\hat {\Omega }(u_s,u_t)\,\rmd s\wedge \rmd t \qquad \text {and} \qquad \int _{\{R\}\times S^1}\lambda (u_t)\,\rmd t \]

for \(R=-\ln \varepsilon \). By the Sobolev inequality the \(C^1\)-norm of \(u\,\rme ^{\sigma s}\) on \([0,\infty )\times S^1\) is bounded by \(\|u\|_{3,\sigma }\), so that up to a positive constant the absolute value of the integrals is bounded by

\[ \|u\|_{3,\sigma }^2\int _R^{\infty }\rme ^{-2\sigma s}\rmd s \qquad \text {and} \qquad \|u\|_{3,\sigma }\,\rme ^{-\sigma R} \,, \]

resp. In both cases the first factor is bounded by assumption; the second tends to zero for \(R\ra \infty \) and the claim follows, namely, that \(\int _Su^*\hat {\Omega }\) is well defined.

We call \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\) a nodal disc map provided that the following conditions are satisfied (cf. Section 5.3 ):

1. \(u\in H^{3,\sigma }(S,j)\),
2. the symplectic energy integral restricted to a connected component \(C\) of \(S\)

   \[ \int _Cu^*\hat {\Omega }\geq 0 \]

   is non-negative for all spherical components \(C\) of \(S\); positive on the disc component,

3. the continuous map on \(S/D\) induced by \(u\) is homologous to a local Bishop discs \(u_{\varepsilon ,b_o}\) relative \(N^*\), so that \([u(S)]=[u_{\varepsilon ,b_o}(\D )]\) in \(H_2(\hat {W},N^*)\), and
4. \(u(m_0)\in \gamma \) and \(\vartheta \circ u(m_k)=\rmi ^k\) for \(k=1,2\).

For given \(\big (S,j,D,\{m_0,m_1,m_2\}\big )\) the space

\[ \HH ^{3,\sigma }(S,j) \]

of nodal disc maps is called the space of admissible maps .

It follows that the degree of the \(C^1\)-map \(\vartheta \circ u\co \partial S\ra S^1\) equals \(1\) for all nodal disc maps \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\). With the properties of the symplectic energy integral discussed above we obtain as in item (2) of Section 4.4 that

\[ \int _Su^*\hat {\Omega }= \int _{\partial S} u^*f\cdot (\vartheta \circ u)^*\rmd \theta \,, \]

where \(f\) is a smooth function on \(N\) that is positive on \(N^*\) and vanishes on \(B\cup \partial N\). As \(u\) takes values in \(N^*\) along the boundary \(\partial S\) we get that

\[ \int _Su^*\hat {\Omega } \in \big (0,2\pi \max f\big ] \]

for all nodal disc maps \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\). By non-negativity of the symplectic energy integral on each connected component \(C\) of \(S\) we get that \(\int _Cu^*\hat {\Omega }\) takes values in \(\big [0,2\pi \max f\big ]\). Moreover, as \(\int _Cu^*\hat {\Omega }\) only depends on the homology class represented by \(u(C)\) for the spherical components \(C\) of \(S\) assumption (2) puts an open condition to the space defined via \(H^{3,\sigma }(S,j)\) and the constraints given by (3) and (4), so that the space of admissible maps \(\HH ^{3,\sigma }(S,j)\) is an open subset.

In fact, \(\HH ^{3,\sigma }(S,j)\) is a Hilbert manifold whose tangent space \(\HH ^{3,\sigma }\big (u^*T\hat {W}\big )\) at \(u\in \HH ^{3,\sigma }(S,j)\) is the space of \(H^{3,\sigma }\)-sections into \(u^*T\hat {W}\) that descent to continuous sections on \(S/D\), that are tangent to \(N^*\) along \(\partial S\) as well as tangent to \(\gamma \) at \(m_0\) and to the page \(\vartheta ^{-1}(\rmi ^k)\) at \(m_k\) for \(k=1,2\). This follows with the exponential map taken w.r.t. a metric on \(\hat {W}\) for which each of the submanifolds \(N\), \(\vartheta ^{-1}(\rmi ^k)\), \(k=1,2\), and \(\gamma \) is totally geodesic. The requirement for the sections to be of class \(H^{3,\sigma }\) is understood as in Section 7.1 , so that a norm on \(\HH ^{3,\sigma }\big (u^*T\hat {W}\big )\) as on [43], p. 66 can be defined. This turns \(\HH ^{3,\sigma }(S,j)\) into a Riemannian Hilbert manifold.

By removal of singularities (see [54] ) a holomorphic \(u\in H^{3,\sigma }(S,j)\), which is continuous and has finite symplectic energy by the above discussion, is holomorphic on \(S\). Therefore, \(u\) is smooth up to the boundary including all nodal points \(|D|\) so that holomorphicity coincides with the notion of holomorphicity from the beginning of this section.

Given a nodal disc map \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\) we call a connected component \(C\) of \(S\) with vanishing symplectic energy integral a ghost bubble . Observe that a holomorphic nodal disc map restricted to a ghost bubble is constant. If \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\) is any nodal disc map such that each ghost bubble admits at least \(3\) nodal points, then we call \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\) a stable nodal disc map .

We call two stable nodal disc maps

\[ \big (S,j,D,\{m_0,m_1,m_2\},u\big ) \quad \text {and} \quad \big (S',j',D',\{m'_0,m'_1,m'_2\},u'\big ) \]

equivalent if there exists a diffeomorphism \(\varphi \co S\ra S'\) such that \(\varphi ^*j'=j\), the injection \(D\ra D'\) defined by \(\{\varphi (x),\varphi (y)\}\in D'\) for all \(\{x,y\}\in D\) is surjective, \(\varphi (m_k)=m'_k\) for \(k=0,1,2\), and \(u'\circ \varphi =u\). The discussions in Section 7.1 about \(H^{3,\sigma }(S,j)\) imply that this equivalence relation is well defined. The equivalence classes

\[ \mathbf u=\big [S,j,D,\{m_0,m_1,m_2\},u\big ] \]

are called stable nodal discs in \(\big (\hat {W},\hat {\Omega }\big )\) relative \(N^*\). The space of all equivalence classes is denoted by \(\ZZ \).

Fixing the diffeomorphism type of \(S\) and the combinatorial data \(\big (D,\{1,\rmi ,-1\}\big )\) of \(\big (S,j,D,\{1,\rmi ,-1\},u\big )\) as at the beginning of Section 6.3 we can write

\[ \ZZ =\Big \{ \bfu =[j,u]\,\,\text {stable} \,\Big |\, [u(S)]=[u_{\varepsilon ,b_o}(\D )] \,,\,\, u(1)\in \gamma \,,\,\, \vartheta \circ u(\rmi ^k)=\rmi ^k \,, k=1,2 \Big \} \]

for the space of all stable nodal discs

\[ \bfu =\big [S,j,D,\{1,\rmi ,-1\},u\big ]\equiv [j,u] \,,\quad u\in \HH ^{3,\sigma }(S,j)\,, \]

in \(\big (\hat {W},\hat {\Omega }\big )\) relative \(N^*\).

We define the nodal type \(\tau \) of \(\big (S,D,\{1,\rmi ,-1\}\big )\) as in Section 6.2 . Namely, the nodal type is the isomorphism class of the rooted tree given as follows: The vertices correspond to the components of \(S\). The root is given by the disc component. The edge relation is induced by the nodes in \(D\). As this time there are no auxiliary marked points all vertices different from the root are not weighted; the root has weight \(3\). The induced nodal type \(\tau \) is necessarily unstable provided that there is at least one sphere component. Indeed, in this case, any end of a branch admits only one special point.

We denote by \(\ZZ _{\tau }\) the space of all stable nodal discs of nodal type \(\tau \), so that \(\ZZ \) is the disjoint union of the \(\ZZ _{\tau }\) where \(\tau \) ranges over all nodal types just described. Each of the subspaces \(\ZZ _{\tau }\) of \(\ZZ \) is the quotient of the total space of the fibration over \(\JJ \equiv \JJ (S)\) with fibre \(\HH ^{3,\sigma }(S,j)\) over \(j\in \JJ \) by the action \(\varphi \mapsto (\varphi ^*j,u\circ \varphi )\) of the group of orientation preserving diffeomorphisms \(\varphi \) of \(S\) preserving \(\big (D,\{1,\rmi ,-1\}\big )\), cf. Section 6.3 . This puts a topology to \(\ZZ _{\tau }\) similarly to Remark 6.3.2 .

Notice that the stabiliser of the action is finite by the stability condition formulated at the end of Section 7.1 : Each automorphism of \(\bfu \) acts via the identity map on the disc component due to the ordered boundary marked points \(\{1,\rmi ,-1\}\). If \(\int _Cu^*\hat {\Omega }=0\) for a connected component \(C\) of \(S\), then the number of nodal points \(C\cap |D|\) on \(C\) is at least \(3\). Furthermore, the automorphisms of \(\bfu \) preserve those ghost components due to the transformation formula. If \(\int _Cu^*\hat {\Omega }>0\), one finds \(z\in C\setminus |D|\) such that \(u\) is immersive on \(C\cap u^{-1}\big (u(z)\big )\) defining finitely many local branches via \(C\cap u^{-1}\big (B_r\big )\subset C\setminus |D|\) for a sufficiently small ball \(B_r\subset \Int \hat {W}\) around \(u(z)\). In fact, due to the positivity of the symplectic energy integral we can find \(z\in C\setminus |D|\) and \(r>0\) sufficiently small such that \(u^*\hat {\Omega }\) is a positive area form on the branch through \(z\), which is oriented via \(j\). Observe that \(u^*\hat {\Omega }\) is a positive area form on all branches through the points of the orbit (of the automorphism group of \(\bfu \)) defined by \(z\). Identifying the sphere components with \((\C P^1,\rmi )\) as in Remark 6.3.1 the identity theorem yields that an automorphism of \(\bfu \) acts by a permutation on the local branches. This proves finiteness of the stabiliser.

In the following we describe a polyfold structure on \(\ZZ \) that glues the components \(\ZZ _{\tau }\) together. For any \(\bfu =\big [S,j,D,\{1,\rmi ,-1\},u\big ]\) in \(\ZZ \) one can choose a so-called stabilisation , which is a finite set of auxiliary marked points \(A\subset S\) disjoint from the special points \(D\cup \{1,\rmi ,-1\}\) such that the nodal disc \(\big (S,j,D,\{1,\rmi ,-1\},A\big )\) is stable in the sense of Section 6.2 . Due to the stability condition there is no need to provide the ghost bubbles with an auxiliary markt point. In addition one can assume, that the automorphisms of \(\bfu \) preserve \(A\), \(u(A)\) is disjoint from the \(u\)-image of \(D\cup \{1,\rmi ,-1\}\), and the following two conditions hold:

1. Whenever \(z,w\in A\) are mapped to the same point \(u(z)=u(w)\) in \(\hat {W}\), then there exists an automorphism of \(\bfu \) sending \(z\) to \(w\).
2. For all \(z\in A\) the \(2\)-form \((u^*\hat {\Omega })_z\) is positive on \((T_zS,j_z)\).

This follows with [43], Lemma 3.2 ignoring the disc component, which already is stable: Namely, successively select finite orbits of the action of the automorphism group of \(\bfu \) on local branches similarly to the above finiteness argument until all components are stable. Consequently, the underlying stable nodal disc \(\big [S,j,D,\{1,\rmi ,-1\},A\big ]\) possesses a uniformiser as described in Section 6.11 .

As in Section 5.3 we wish to achieve an index-\(1\) Fredholm problem. In view of Theorem 6.11.1 we compensate the stabilising auxiliary marked points \(A\) index-wise as follows: We choose a finite collection of pairwise disjoint codimension-\(2\) symplectic discs in \((\Int \hat {W},\hat {\Omega })\) that intersect \(u(S)\) along \(u(A)\) transversally. This is possible by condition (2) above. Namely, the image of \(Tu\) at each auxiliary marked point in \(A\) is a symplectic plane in \(T\hat {W}\). Integrating the respective symplectic normal subspaces one finds symplectic embeddings of small discs of codimension \(2\) that are normal to \(u(S)\) at the images of the auxiliary marked points \(u(A)\). We call the union of the discs \(H_{u,A}\) local transversal constraints if the intersection of \(u(S)\) and \(H_{u,A}\) equals \(u(A)\) and if each component of \(H_{u,A}\) intersects \(u(S)\) in a single point.

We denote by

\[ E_{u,A}\subset \HH ^{3,\sigma }\big (u^*T\hat {W}\big ) \]

the subspace of sections that are tangent to \(H_{u,A}\) at the stabilising auxiliary points in \(A\), which is scale-linear w.r.t. to \((3+\nu ,\sigma _{\nu })\), \(\nu \in \N _0\), for a strictly increasing sequence \(\sigma _{\nu }\) in \((0,1)\) with \(\sigma _0=\sigma \), see [43], Section 2.6 . Uniformiser about any \(\bfu =\big [S,j,D,\{1,\rmi ,-1\},u\big ]\) in \(\ZZ \) of the desired polyfold structure are obtained as in [43], Section 3.1/3.2 . To adapt to our situation start off with uniformisers for the stabilised domain \(\big [S,j,D,\{1,\rmi ,-1\},A\big ]\in \RR \) from Section 6.11 and consider the deformation

\[ (y,\bfa ,\eta ) \longmapsto \big (S_{\bfa },j(y)_{\bfa },D_{\bfa },\{1,\rmi ,-1\},\oplus _{\bfa }\exp _u(\eta )\big ) \,, \]

where \((y,\bfa )\in \VV \) for an open subset \(\VV \) of \(V_j\times \D ^D\), \(\eta \in E_{u,A}\) is a sufficiently small section that is a fixed point of the splicing projection \(\pi _{\bfa }\) and \(\oplus _{\bfa }\exp _u(\eta )\) denotes the gluing operation both introduced in [43], Section 2.4/2.5 . Choosing \(u\) to be a smooth approximation of an element in \(H^{3,\sigma }(S,j)\) we obtain scale-smooth gluing maps w.r.t. to the scale \((3+\nu ,\sigma _{\nu })\), see [43], Section 2.2/2.6 . Using Remark 6.3.2 , Section 6.11 and [43], Section 3.3/3.4 one obtains a natural second countable paracompact Hausdorff topology on \(\ZZ \) similarly to [43], Theorem 1.6 . In the same way using this time modifications in [43], Section 3.5 the space \(\ZZ \) carries the structure of a polyfold as formulated in [43], Theorem 1.7 with a scale-smooth evaluation map \(\ZZ \ra \gamma \) sending \(\bfu \) to \(u(1)\), cf. [43], Theorem 1.8 .

We call \(\bfu \) a stable nodal holomorphic disc if \(\bfu \) can be represented by a stable nodal holomorphic disc map \(u\). Notice, that all stable nodal disc maps \(u\) that represent a stable nodal holomorphic disc \(\bfu \) are holomorphic. Denote by

\[ \NN :=\big \{ \bfu \in \ZZ \,\big |\, \bfu \,\,\,\text {is holomorphic} \big \} \]

the nodal moduli space of all stable nodal holomorphic discs.

Using uniformisation it is convenient to represent the classes \(\mathbf u\in \NN \) by holomorphic maps \(u\in \HH ^{3,\sigma }(S,j)\) whose disc component has domain \(\big (\D ,\rmi ,\{1,\rmi ,-1\}\big )\) and for which the sphere components are given by \((\C P^1,\rmi )\), cf. Remark 6.3.1 . If \(\bfu \) is un-noded, then we obtain \(\bfu =\big [\D ,\rmi ,\emptyset ,\{1,\rmi ,-1\},u\big ]\). We abbriviate the elements \(\mathbf u=[\rmi ,u]\in \NN \) (noded or un-noded) simply by \([u]\) for the following discussion:

The boundary conditions for \(\HH ^{3,\sigma }(S,j)\) formulated in Section 7.1 are the boundary conditions used in Sections 4.4 and 5.3 . In particular, all properties formulated in the un-noded case for holomorphic discs in Section 4.4 continue to hold in the noded case, hence, for all \(\bfu =[u]\in \NN \) in the following sense:

1. The winding number of \(\bfu =[u]\in \NN \), which by definition is the degree of the map \(\vartheta \circ u\co \partial S\ra S^1\), is equal to \(1\). In particular, \(u(\partial S)\) is an embedded curve in \(N^*\) positively transverse to \(\xi \) and the restriction of \(u\) to the disc component of \(S\) is a simple holomorphic map.
2. The symplectic energy \(\int _{S}u^*\hat {\Omega }\) of \(\bfu =[u]\in \NN \), which is well defined and positive by Section 7.1 , is uniformly bounded.
3. The boundary circle \(u(\partial S)\) of \(\bfu =[u]\in \NN \) is disjoint from \(U_{\partial N}\) because the restriction of \(u\) to the disc component of \(S\) must be disjoint from \(U_{\partial N}\) by Lemma 3.5.1 . If \(u(S)\) intersects \(U_B\) then \(\bfu \) is un-noded and equivalent to a local Bishop disc \(u_{\varepsilon ,b_o}\) by Lemma 3.3.1 combined with the final paragraph of Remark 4.4.1 .

The local Bishop discs \(u_{\varepsilon ,b_o}\), \(\varepsilon \in (0,\delta )\), represent elements

\[ \bfu _{\varepsilon ,b_o}= \big [\D ,\rmi ,\emptyset ,\{1,\rmi ,-1\},u_{\varepsilon ,b_o}\big ] \]

in \(\NN \). The corresponding local Bishop filling can be identified with \((0,\delta )\). We truncate the nodal moduli space \(\NN \) via

\[ \NN _{\cut }=\NN \setminus (0,\delta /2)\,. \]

The moduli space \(\NN \) is the zero set

\[ \NN =\big \{ \bfu \in \ZZ \,\big |\, \CR {\!\hat {J}}\mathbf u=\mathbf 0 \big \} \]

of the Cauchy–Riemann operator \(\CR {\!\hat {J}}\), which appears as a section into the bundle

\[ p\co \WW \lra \ZZ \]

over \(\ZZ \). The fibre of \(p\) over \(\mathbf u=\big [S,j,D,\{1,\rmi ,-1\},u\big ]\in \ZZ \) consists of equivalence classes \(\boldxi =\big [S,j,D,\{1,\rmi ,-1\},u,\xi \big ]\) of continuous sections \(\xi \) of \(\Hom \big (TS,u^*T\hat {W}\big )\) so that for each \(z\in S\) the map \(\xi (z)\co T_zS\ra T_{u(z)}\hat {W}\) is complex anti-linear with respect to \(j(z)\) and \(\hat {J}\big (u(z)\big )\). Moreover, \(\xi \) is of Sobolev class \(H^2_{\loc }\) on \(S\setminus |D|\) and of weighted Sobolev class \(H^{2,\sigma }\) near \(|D|\) similarly to the description at the beginning of Section 7.1 , see [43], Section 1.2 . Two such sections \(\big (S,j,D,\{m_0,m_1,m_2\},u,\xi \big )\) and \(\big (S',j',D',\{m'_0,m'_1,m'_2\},u',\xi '\big )\) are equivalent , if there exists an equivalence \(\varphi \) of stable nodal disc maps \(\big (S,j,D,\{m_0,m_1,m_2\},u\big )\) and \(\big (S',j',D',\{m'_0,m'_1,m'_2\},u'\big )\) as described at the beginning of Section 7.2 such that \(\xi '\circ T\varphi =\xi \). By adapting [43], Theorem 1.9 to the situation of the current Sections 6 and 7 we obtain a natural second countable paracompact Hausdorff topology on the total space \(\WW \) and the bundle projection \(p\co \WW \ra \ZZ \) that maps \(\big [S,j,D,\{1,\rmi ,-1\},u,\xi \big ]\) to \(\big [S,j,D,\{1,\rmi ,-1\},u\big ]\) is continuous. Furthermore \(p\co \WW \ra \ZZ \) constitutes a strong polyfold bundle in view of [43], Theorem 1.10 .

The Cauchy–Riemann operator \(\CR {\!\hat {J}}\) is the section of \(p\) given by

\[ \CR {\!\hat {J}}\mathbf u := \Big [ S,j,D,\{1,\rmi ,-1\},u,\tfrac 12\big (Tu+\hat {J}(u)\circ Tu\circ j\big ) \Big ] \]

for all \(\mathbf u=\big [S,j,D,\{1,\rmi ,-1\},u\big ]\in \ZZ \). For a representative we write \(\CR {\!\hat {J}}u\) also. As in [43], Theorem 1.11 the Cauchy–Riemann operator \(\CR {\!\hat {J}}\co \ZZ \ra \WW \) is a scale-smooth component-proper Fredholm section that admits a natural orientation which we describe in Remark 7.4.3 below. The Fredholm index of \(\CR {\!\hat {J}}\co \ZZ \ra \WW \) is \(1\) by the index computation in Section 5.3 taking local transversal constraints from Section 7.2 in view of Theorem 6.11.1 into account. As in [64], Section 5.3 the vertical differential of a local representation of \(\CR {\!\hat {J}}\) near the local Bishop discs \(\bfu _{\varepsilon ,b_o}\), \(\varepsilon \in (0,\delta )\) has a right-inverse. The same holds true for all simple stable nodal holomorphic discs in \(\ZZ \) due to the generic choice of \(\hat {J}\), see Section 5.3 .

Proof of Theorem 5.1.2 part (ii).