Teardrop curves and obstructions [04H9]

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Teardrop curves and obstructions

The assumptions on $(X,\omega)$ are as in the previous section. Immersed Lagrangians are immersions $\iota:L\to X$ with $\omega|_{L}=0$ , and all self intersections are transverse. The domain of $L$ is allowed to be disconnected, so the union of finitely many transversely intersecting embedded Lagrangians are examples of immersed Lagrangians. Each self intersection point of two local sheets $L_{+},L_{-}$ corresponds to two different points $x_{\pm}$ on the domain of $L$ . It is important to distinguish $x_{+}$ and $x_{-}$ , because for the boundary of the holomorphic curve to pass through $x_{+}$ in the clockwise direction means crossing from $L_{+}$ to $L_{-}$ , and $x_{-}$ signifies the opposite crossing.

We say $L$ is exact, if there is a function $f_{L}$ on the domain of $L$ , such that $df_{L}$ agrees with the Liouville 1-form restricted to $L$ . For energy reasons, this forbids nontrivial holomorphic disks with boundary on $L$ which never change local sheets at any boundary point. The caveat is that the relative homology class $[\omega]\in H_{2}(X,L)$ may still be nonzero. The brane structures on $L$ are as in the embedded case. The construction of $CF^{*}(L,L)$ depends on the approach, but a common feature is that it includes

CF^{*}_{self}(L,L)=\bigoplus_{\text{self intersection $p$}}CF^{*}(L_{+},L_{-})\oplus CF^{*}(L_{-},L_{+}).

generated by the local system factor $\text{Hom}(E_{+},E_{-})|_{p}$ (resp. $\text{Hom}(E_{-},E_{+})|_{p}$ ) tensored with the orientation line.

The Gromov compactness discussion is largely similar to the embedded case. A new phenomenon is the teardrop curves, namely the holomorphic curves with boundary on $L$ and a single output corner at a self intersection point $r\in CF^{*}(L_{+},L_{-})$ . Of particular importance is the case with $\mu_{L_{+},L_{-}}(r)=2$ . The number $2$ is intuitively explained by the 2 degrees of freedom of the domain Möbius transforms fixing the corner point $Aut(D^{2},1)$ , modulo which such teardrop curves occur in dimension zero moduli spaces.

Now if we attempt to run the usual argument for $d^{2}=0$ in Floer cohomology, we would consider the moduli space of holomorphic strips between $p,q$ with $\deg q-\deg p=2$ , modulo the translation $\mathbb{R}$ . However, in addition to the usual strip breaking, the holomorphic strips can also break into a holomorphic triangle with inputs $p,r$ and output $q$ , and a teardrop curve with corner at $r$ . In summary, teardrop curves with corner at a degree 2 intersection point obstruct Floer cohomology.

The automorphism group $Aut(D^{2},1)$ forbids the naïve domain dependent perturbation schemes, which in turn causes transversality problems. In the literature there are two approaches to solve this problem: Joyce and Akaho [7] use virtual perturbation techniques for bordered Riemann surfaces, while Woodward et al. [81][82] circumvent the virtual perturbations by utilizing stabilising divisors. Both approaches assign curved $A_{\infty}$ algebra structures $(m_{0},m_{1},\ldots)$ to the Floer cochain spaces $CF^{*}(L,L)$ of immersed Lagrangians. In the exact setting, the $m_{0}\in CF_{self}^{2}(L,L)$ term amounts to a count of teardrop curves with corner at degree 2 self intersection points, with weighting factors coming from the holonomy of the local system. Since in the main text the emphasis is on the automatic transversality assumption, we shall not dwell on the details of perturbation schemes, but only identify a few simplifications in the exact setting.

Remark 6.11.

The rough idea of Woodward et al. is to introduce interior marked points, constrained to lie on a Donaldson divisor $D$ disjoint from the Lagrangians. The virtual dimension is not affected by these divisor constraints, since each interior marked point increases it by 2, while each divisor constraint decreases it by 2. One needs to arrange $D$ to be of sufficiently high degree, so that each nontrivial pseudoholomorphic disk with boundary on the Lagrangians has at least one intersection with $D$ . On a teardrop curve, imposing the divisor constraint at interior marked points kills the domain automorphisms $Aut(D^{2},1)$ , so one can then introduce domain dependent perturbation of almost complex structures compatible with $D$ to achieve sufficient transversality to make sense of counts. The appealing feature of this approach, is that adding marked points does not alter the geometric interpretation of the holomorphic curves, so stays closer to geometry than the virtual approach.

The framework of Woodward et al. [81][82] is not restricted to exact settings, and works also for compact symplectic manifolds with rational $[\omega]\in H^{2}(X)$ . Producing the Donaldson divisor with the intersection properties is easier if $[\omega]\in H_{2}(X,L)$ is a rational class, although the methods in [14, section 3.1] allows one to largely relax this assumption.

In exact manifolds, as mentioned in [81, Remark 4.5], one can avoid the spherical components of the treed disks. In the exact Lagrangian setting, the only bubbling happens at the self intersection points. These afford significant simplifications to the construction, and allows one to think of the treed disks in [14][15] [81][82] in terms of a tree of holomorphic polygons connected at the self intersection points. By avoiding the troublesome sphere bubbles, one can also relax the restriction of moduli spaces of dimension at most one.

Remark 6.12.

A very technical aspect of Akaho-Joyce [7] is that the $A_{\infty}$ structure is not constructed directly, but through a sequence of approximations involving energy cutoff scales. In the exact setting, the topological energy formula implies a priori energy bounds, so this complication would not arise.