What if we relax the positivity condition? [04BH]

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What if we relax the positivity condition?

We now discuss a weaker version that does not require the positivity condition on holomorphic curves. This amounts to dropping pointwise positivity of the integrand for the moduli space integral, which breaks some parts of our mirror analogy.

As in Theorem 3.21, we consider $L^{\prime}$ built from two immersed Lagrangians $L_{1},L_{2}$ , and $L$ fits into the distinguished triangle. All Lagrangians are almost calibrated, and all intersections are transverse. We classify the automatically transverse holomorphic curves (cf. Cor. 3.6, Prop. 3.8) into $\pm$ types according to whether the boundary evalation to $L$ agrees with the orientation on $L$ or its opposite. The complex volume forms $\tilde{\Omega}_{L_{i}}$ on the moduli space can be split into the sum of two parts according to whether $u:\Sigma\to X$ is of $\pm$ types:

\tilde{\Omega}_{L_{i}}=\tilde{\Omega}_{L_{i}}^{+}+\tilde{\Omega}_{L_{i}}^{-},\quad\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}=\int_{L_{i}}\Omega.

In particular the signed measure $\text{Re}\tilde{\Omega}_{L_{i}}$ is decomposed into its positive and negative parts, and $\text{Re}\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{-}\leq 0$ . We define

\hat{\theta}_{i}^{+}=\arg\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{+},\quad\hat{\theta}_{i}^{-}=\arg(-\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{-}),\quad i=1,2.

Here $\hat{\theta}_{i}^{-}$ is only defined when $\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{-}$ is nonzero, namely the case not covered already by the positivity condition.

Theorem 3.28.

(Floer theoretic obstruction, relaxing positivity condition) Assume the automatic transversality holds for the bordism current $\mathcal{C}$ between $L$ and $L^{\prime}$ . Then the Lagrangian phase angle of $L$ has a lower bound on its oscillation:

\sup_{L}\theta_{L}\geq\max\{\hat{\theta}_{1}^{+},\hat{\theta}_{1}^{-}\},\quad\inf_{L}\theta_{L}\leq\min\{\hat{\theta}_{2}^{+},\hat{\theta}_{2}^{-}\}.

Proof.

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

Claim 3.29.

Clockwise along $\partial\Sigma$ , the function $\text{Re }F$ is decreasing on the $L$ boundary portion, but increasing on the $L^{\prime}=L_{1}\cup L_{2}$ boundary portion. In particular,

0=\text{Re }F(q)\geq\text{Re }F\geq\text{Re }F(p).

More intrinsically, the real part of the complex volume forms on the moduli spaces at such $u:\Sigma\to X$ are nonpositive.

The corresponding claim 3.23 is modified to

Claim 3.30.

The image $F(\Sigma)\subset\mathbb{C}$ lies below its $L^{\prime}$ boundary portion, and above its $L$ boundary portion.

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,

\begin{split}&\text{Re}\int_{L}\chi_{A_{i}^{+}}\Omega=\text{Re}\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{+}\geq 0,\quad\text{Re}\int_{L}\chi_{A_{i}^{-}}\Omega=\text{Re}\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}^{-}\leq 0,\\ &\text{Im}\int_{L}\chi_{A_{2}^{+}}\Omega\leq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{2}}^{+},\quad\text{Im}\int_{L}\chi_{A_{1}^{+}}\Omega\geq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{1}}^{+},\\ &\text{Im}\int_{L}\chi_{A_{2}^{-}}\Omega\geq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{2}}^{-},\quad\text{Im}\int_{L}\chi_{A_{1}^{-}}\Omega\leq\text{Im}\int_{\mathcal{M}}\tilde{\Omega}_{L_{1}}^{-}.\end{split}

In particular

\arg\int_{L}\chi_{A_{2}^{+}}\Omega\leq\hat{\theta}_{2}^{+},\quad\arg(-\int_{L}\chi_{A_{2}^{-}}\Omega)\leq\hat{\theta}_{2}^{-},

and

\arg\int_{L}\chi_{A_{1}^{+}}\Omega\geq\hat{\theta}_{1}^{+},\quad\arg(-\int_{L}\chi_{A_{1}^{-}}\Omega)\geq\hat{\theta}_{1}^{-}.

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. ∎

Remark 3.12.

In the special case where $L_{1},L_{2}$ are special Lagrangians of phase $\hat{\theta}_{1},\hat{\theta}_{2}$ , then clearly $\hat{\theta}_{i}^{\pm}=\hat{\theta}_{i}$ . The conclusion in this case can be deduced easily from Floer degree considerations at Lagrangian intersections, similar to section 2.2.

Remark 3.13.

Recall $\hat{\theta}_{i}=\arg\int_{L_{i}}\Omega$ . The caveat is that $\max\{\hat{\theta}_{i}^{\pm}\}$ and $\min\{\hat{\theta}_{i}^{\pm}\}$ do not quite control $\hat{\theta}_{i}$ , so the above Theorem 3.28 does not imply the phase angle inequality (31). In this sense the conclusion of Theorem 3.28 is weaker than Theorem 3.21, illustrating the power of the positivity condition.

On the other hand, we will heuristically argue in section 3.6 that in the Thomas-Yau-Joyce picture, once we assume the existence of Joyce’s Bridgeland stability condition, the phase angle inequality (31) can be deduced without the positivity condition in Theorem 3.21.

We think it is very interesting to either prove the positivity condition as a consequence of the other assumptions, or to find another Floer theoretic argument for (31) that requires neither the positivity condition, nor the a priori knowledge of special Lagrangian representatives.