At a holomorphic polygon $\Sigma$ in the universal family $\mathcal{C}$, denote $v_{1},\ldots v_{n-1}$ as the first order deformation vector fields representing an oriented basis of tangent vectors to the moduli space. In the special case of holomorphic strips, the moduli space refers to the $\mathbb{R}$-translation quotient. We noted in section 3.3 that $\Omega(\cdot,v_{1},\ldots,v_{n-1})$ restricts to a holomorphic 1-form on $\Sigma$, so can be written as the differential of a holomorphic function $F$ by the simply connectedness of $\Sigma$:

The corners on $\Sigma$ are arranged in clockwise order with the following possibilities:

• •

  In the primary case, we encounter some degree one self intersections on $L$ from bounding cochains, a corner $p\in CF^{0}(L,L_{2})$, some degree one self intersections on $L_{2}$, a corner $r$ from $\gamma\in CF^{1}(L_{2},L_{1})$, some degree one self intersections on $L_{1}$, and a corner at $q\in CF^{0}(L_{1},L)$. Notice the Lagrangian boundary follows $L,L_{2},L_{1}$ in clockwise order, and we cannot go reversely from $L_{1}$ to $L_{2}$ instead.

• •

  In the secondary cases, the boundary data may miss either $L_{1}$ or $L_{2}$. For instance, we may encounter some degree one intersections on $L$, a corner $p\in CF^{0}(L,L_{2})$, some degree one intersections on $L_{2}$ and a corner at $q\in CF^{0}(L_{2},L)$. The Lagrangian boundary follows $L,L_{2}$ in clockwise order. The alternative possibility of Lagrangian boundary along $L$ and $L_{1}$ is entirely similar.

In all cases, there is precisely one corner $p$ at $CF^{0}(L,L^{\prime})$ and a corner $q$ at $CF^{0}(L^{\prime},L)$. We can normalize $F(q)=0$ to fix the constant. In the primary case, there is a corner $r\in CF^{1}(L_{2},L_{1})$, which is absent in the secondary cases. In general, the bordism current $\mathcal{C}$ receives contributions from many moduli spaces, and all three cases may arise depending on the generators of $HF^{0}(L,L^{\prime})$ and $HF^{0}(L^{\prime},L)$.

We can now define complex valued volume forms on the $(n-1)$ dimensional moduli spaces of holomorphic curves. Recall $v_{1},\ldots v_{n-1}$ represent the tangent vectors to the moduli spaces, and the holomorphic function $F$ depends on $v_{1}\wedge\ldots v_{n-1}$. In the primary case, we define

In the secondary cases, if the Lagrangian boundary lies on $L$ and $L_{1}$, then

If the Lagrangian boundary lies on $L$ and $L_{2}$, then

The values of $F$ should be understood as the integral of $dF$ on the appropriate portions of $\partial\Sigma$. The key point is that since $\partial\mathcal{C}$ sweeps out the cycle $L-L_{1}-L_{2}$, we can write the period integrals as integrals on the $(n-1)$-dimensional moduli spaces of holomorphic curves:

where $\mathcal{M}$ is a shorthand for the weighted sum over contributions from all the $(n-1)$-dimensional moduli spaces involved in the construction of $\mathcal{C}$, cf. the Appendex 6.2.

Recall the positivity condition means that if $v_{0}$ stands for a clockwise oriented tangent vector on $\partial\Sigma$, then $v_{0}\wedge v_{1}\ldots\wedge v_{n-1}$ agrees with the orientation on $L$, and is opposite to the orientation on $L^{\prime}$. The nonvanishing of $v_{0}\wedge v_{1}\ldots\wedge v_{n-1}$ is a consequence of the immersion property from the automatic transversality (cf. Cor. 3.6, Prop. 3.8). The almost calibrated condition implies that $\text{Re}\Omega>0$ on the Lagrangians with respect to the orientation on $L$ and $L^{\prime}$. Thus

The holomorphic function $F$ maps $\Sigma$ into a bounded region in the complex plane. The behaviour at the corners is specified in Remark 3.7. Since each vertical line intersects $\partial F(\Sigma)\subset\mathbb{C}$ at $\leq 2$ points by the monotonicity claim above, the boundary and corner local behaviours imply that

We turn to the proof of the Lagrangian phase angle inequality (31). For each curve that contributes nontrivially to $\tilde{\Omega}_{L}$, by the monotonicity claim we can find a unique point $r^{\prime}$ on the $L$ boundary of $\partial\Sigma$, such that

From the image curve claim, we always have $\text{Im}F(r^{\prime})\geq\text{Im}F(r)$ in the primary case. Integrating over the moduli space of holomorphic curves,

We now introduce two almost everywhere defined functions $\chi_{A_{1}},\chi_{A_{2}}$ on $L$. The recipe is that at any generic point $P\in L$, if an automatically transverse holomorphic curve in the universal family passes through $P$ on the boundary portion of $\partial\Sigma$ joining $q$ to $r^{\prime}$ (resp. $r^{\prime}$ to $p$), then it gives an additive contribution to $\chi_{A_{1}}(P)$ (resp. $\chi_{A_{2}}(P)$) equal to the weighting factor of the curve. Intuitively $\chi_{A_{1}},\chi_{A_{2}}$ should be understood as the characteristic functions of weighted subsets $A_{1},A_{2}\subset L$. The positivity condition gives $\chi_{A_{i}}\geq 0$, and $\partial\mathcal{C}=L-L^{\prime}$ gives $\chi_{A_{1}}+\chi_{A_{2}}=1$. Intuitively $A_{1},A_{2}$ give a (weighted) partition of $L$.

The moduli space integrals now have target space interpretations:

Since $L$ is homologous to $L_{1}+L_{2}$, we have $\int_{L}\Omega=\int_{L_{1}}\Omega+\int_{L_{2}}\Omega$. Whence

Finally we deal with the J-volume lower bound (32). By the triangle inequality,

The RHS is at least $|\int_{L_{1}}\Omega|+|\int_{L_{2}}\Omega|$, due to an elementary numerical fact:

This concludes the proof of (32). ∎

Since most parts of the proof are identical to the distinguished triangle case, we will only sketch the main difference.

The Lagrangian boundaries on $\partial\Sigma$ are arranged in the clockwise order as $L,L_{N},L_{N-1},\ldots L_{1}$. We construct the holomorphic function $F$ as in (33), and use it to produce complex valued volume forms on the $(n-1)$-dimensional moduli spaces, such that for any $m=1,\ldots N$,

As before, the real part of these complex volume forms are all non-negative, as a consequence of the positivity condition. The claim on the image curve $F(\Sigma)$ holds verbatim. Similarly to the distinguished triangle case, we produce nonnegatively weighted subsets $A_{1},\ldots A_{N}\subset L$ with $L=\sum A_{i}$, such that

This implies

whence the phase inequality (35).

The J-volume can be bounded below by

The last step uses the purely numerical Lemma 3.27 below. ∎

We argue by induction. The $N=2$ case is implied by Lemma 3.25. In general, we view $\sum_{1}^{N}|z_{i}|$ as a function of the imaginary parts of $z_{1},\ldots z_{N}$ subject to the constraints. Clearly this function achieves its minimum for some $(z_{i})$. If $z_{1}=a_{1}$, then we can conclude by induction. Otherwise $\text{Im}z_{1}>\text{Im}a_{1}$. If $\arg z_{2}<\arg z_{1}$, then we can fix $z_{1}+z_{2}$ and decrease $\sum_{1}^{2}|z_{i}|$ by Lemma 3.25, which would contradict minimality. Proceding with this argument, we are forced to have

whence

which contradicts $\sum_{1}^{N}z_{i}=\sum_{1}^{N}a_{i}$. ∎

We will only sketch the modifications. The positivity conditition enters through the monotonicity claim 3.22. Once we drop this, we would allow holomophic discs $u:\Sigma\to X$ that sweep out parts of $L\cup L^{\prime}$ with the reversed orientation. For such curves, claim 3.22 is modified to

The corresponding claim 3.23 is modified to

At almost every point on $L\cup L^{\prime}$, only automatically transverse holomophic curves pass through it, since by assumption the boundary evaluation of the other holomorphic curves is contained in some subset of $L\cup L^{\prime}$ with Hausdorff dimension $\leq n-1$. According to the $\pm$ types of the automatically transverse curves, we decompose the weighted characteristic functions $\chi_{A_{i}}$ on $L$ into its positive and negative parts $\chi_{A_{i}^{+}}\geq 0$ and $\chi_{A_{i}^{-}}\leq 0$.

Then upon integration over the moduli space,

In particular

and

Now $\chi_{A_{i}^{+}}\geq 0$ and $\chi_{A_{i}^{-}}\leq 0$, and the special case where $\chi_{A_{i}^{-}}=0$ almost everywhere is already covered by the positivity condition. The Theorem follows. ∎