3.6 Including singular Lagrangians in D b ℱ ( M ) ; LMCF for Lagrangians with stable conical singularities [03P9]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

3.6 Including singular Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$ ; LMCF for Lagrangians with stable conical singularities

The programme of §3.2 involves flows $\{L^{t}:t\in[0,\infty)\}$ with the $L^{t}$ immersed Lagrangians which can be singular at the singular times $t=T_{1},T_{2},\ldots,$ where we do not require $(L^{T_{i}},E^{T_{i}},b^{T_{i}})$ to be objects of $D^{b}{\mathbin{\mathscr{F}}}(M)$ . In this section we argue that in dimension $m\geqslant 3$ , we must also allow the $L^{t}$ to have certain kinds of ‘stable’ singularities for $t\neq T_{i}$ . To complete the programme, Lagrangian MCF must work for such singular Lagrangians, and we must include them as objects in the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ .

In [32, 33, 34, 35, 36] the author studied compact SL $m$ -folds $L$ with isolated conical singularities in a Calabi–Yau $m$ -fold $M$ . That is, $L$ has singularities $p_{1},\ldots,p_{k}$ locally modelled on closed special Lagrangian cones $C_{1},\ldots,C_{k}$ in ${\mathbin{\mathbb{C}}}^{m}$ which have isolated singularities at $0\in{\mathbin{\mathbb{C}}}^{m}$ . As in [33], the deformation theory of $L$ involves an obstruction space ${\mathbin{\cal O}}={\mathbin{\cal O}}_{1}\oplus\cdots\oplus{\mathbin{\cal O}}_{k}$ which is the sum of contributions ${\mathbin{\cal O}}_{i}$ from each singular point $p_{i}$ , depending only on the cone $C_{i}$ . We call the singularities $p_{i}$ and the SL cones $C_{i}$ stable [33, Def. 3.6] if the obstruction spaces ${\mathbin{\cal O}}_{i}$ are zero. By [33, Cor. 6.11], if $L$ has only stable isolated conical singularities, then the moduli space ${\mathbin{\cal M}}_{L}$ of SL deformations of $L$ is a smooth manifold.

Few examples of stable SL cones are known. The SL $T^{2}$ -cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ in equation (2.4) of Example 2.7 was shown to be stable in [32, §3.2]. Ohnita [59] found four more examples of stable SL cones in dimensions 5, 8, 14, and 26. In dimension $m=2$ , any irreducible, immersed SL cone in ${\mathbin{\mathbb{C}}}^{2}$ is a Lagrangian plane ${\mathbin{\mathbb{R}}}^{2}$ , or a finite cover of ${\mathbin{\mathbb{R}}}^{2}$ branched at 0. Nontrivial branched covers of ${\mathbin{\mathbb{R}}}^{2}$ are unstable. So there are no singular stable SL cones in ${\mathbin{\mathbb{C}}}^{2}$ .

Principle 3.18.

(a) In the programme of §3.2, in dimension $m\geqslant 3,$ for the Lagrangians $L^{t}$ at nonsingular times $t\neq T_{i}$ we should allow Lagrangians with ‘stable special Lagrangian singularities’. These should include stable isolated conical singularities, as in [33], and probably also other classes of non-isolated or non-conical singularities.

For example, if $m=k+l$ with $k,l>0$ and $C$ is a stable special Lagrangian cone in ${\mathbin{\mathbb{C}}}^{k}$ as above, the author expects that Lagrangians $L$ with $l$ -dimensional singularities locally modelled on $C\times{\mathbin{\mathbb{R}}}^{l}$ in ${\mathbin{\mathbb{C}}}^{k}\times{\mathbin{\mathbb{C}}}^{l}={\mathbin{\mathbb{C}}}^{m}$ are ‘stable’.

In dimension $m=3,$ Lagrangians with conical singularities modelled on the $T^{2}$ -cone $C$ in (2.4) may be the only kind required. As $m$ increases, the singularities allowed will probably become more and more complicated.

(b) For each such class of stable singularities one should prove short time existence for Lagrangian MCF.

(c) One should extend the definitions of Lagrangian Floer cohomology, obstructions to $HF^{*},$ and $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include each such class of stable singularities.

For (b), the author’s PhD student Tapio Behrndt proved [9, Th. 5.12]:

Theorem 3.19.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$ -fold, and $L$ a compact Lagrangian $m$ -fold in $M$ with isolated conical singularities modelled on stable SL cones in ${\mathbin{\mathbb{C}}}^{m}$ (with any phase $e^{i\phi}$ ). Then for small $\epsilon>0$ there exists a unique smooth family $\{L^{t}:t\in[0,\epsilon)\}$ satisfying Lagrangian MCF with $L^{0}=L,$ where the $L^{t}$ are compact Lagrangians in $M$ with stable isolated conical singularities.

Problem 3.20.

Extend the theories of Lagrangian Floer cohomology, obstructions to $HF^{*},$ and Fukaya categories $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include Lagrangians $L$ in $M$ with isolated conical singularities modelled on stable special Lagrangian cones $C$ in ${\mathbin{\mathbb{C}}}^{m},$ such as the $T^{2}$ -cone $C$ in ${\mathbin{\mathbb{C}}}^{3}$ in (2.4). The main technical issues will involve studying moduli spaces of $J$ -holomorphic discs $\Sigma$ in $M$ whose boundaries $\partial\Sigma$ lie in $L$ and pass through singular points of $L$ .

Problem 3.20 can be approached as an exercise in Symplectic Field Theory, as in Eliashberg et al. [16]: given $L$ with conical singularities at $p_{1},\ldots,p_{k}$ modelled on stable SL cones $C_{1},\ldots,C_{k}\subset{\mathbin{\mathbb{C}}}^{m}$ , we delete $p_{1},\ldots,p_{k}$ from $L,M$ , and treat $M\setminus\{p_{1},\ldots,p_{k}\}$ as a noncompact symplectic manifold with concave cylindrical ends modelled on ${\mathbin{\cal S}}^{2m-1}\times(-\infty,0)$ , and $L\setminus\{p_{1},\ldots,p_{k}\}$ as a noncompact Lagrangian with cylindrical ends modelled on $\Sigma_{j}\times(-\infty,0)$ for $j=1,\ldots,k$ , where $\Sigma_{j}=C_{j}\cap{\mathbin{\cal S}}^{2m-1}$ is the special Legendrian link of the cone $C_{j}$ .

The reason we need to include Lagrangians with ‘stable singularities’ in the programme of §3.2 is that (the author expects) for $m\geqslant 3$ there should exist examples of flows $\{L^{t}:t\in[0,T)\}$ in nonsingular Lagrangians with a finite time singularity at $t=T$ , such that one can only continue the flow for $t>T$ by using Lagrangians with stable singularities.

Example 2.8 described a continuous family of exact SL 3-folds $N^{t}$ in ${\mathbin{\mathbb{C}}}^{3}$ for $t\in(-\epsilon,\epsilon)$ , such that $N^{t}$ is nonsingular for $t<0$ , and $N^{0}$ has one (non-stable) singular point with tangent cone ${\mathbin{\mathbb{R}}}^{3}\amalg_{\mathbin{\mathbb{R}}}{\mathbin{\mathbb{R}}}^{3}$ , and $N^{t}$ for $t>0$ has two singular points modelled on the stable SL $T^{2}$ -cone of (2.4). By Principles 3.9(a) and 3.11, we should expect there to exist similar examples of Lagrangian MCF $L^{t}:t\in(-\epsilon,\epsilon)$ with surgeries, such that $L^{t}$ is nonsingular for $t<0$ with a finite time singularity at $t=0$ , and $L^{0}$ has one singular point with tangent cone ${\mathbin{\mathbb{R}}}^{3}\amalg_{\mathbin{\mathbb{R}}}{\mathbin{\mathbb{R}}}^{3}$ , and $L^{t}$ has two stable singularities modelled on $C$ in (2.4).

Remark 3.21.

We temporarily write $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ for the derived Fukaya category of nonsingular immersed Lagrangians, and $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ for the category including Lagrangians with ‘stable special Lagrangian singularities’. It seems likely that $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ and $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ need not be equivalent categories. If so, $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ may be preferable to $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ , in the sense of being better behaved, more natural, or the right category to use in Mirror Symmetry. To test this, we should start in dimension $m=3$ by including Lagrangians with isolated singularities modelled on the $T^{2}$ -cone $C$ in (2.4).

The following example was suggested to me by Ivan Smith. Harris [25] constructs a smooth family $(M^{t},\omega^{t}):t\in[0,\epsilon)$ of symplectic Calabi–Yau 6-manifolds for small $\epsilon>0$ , with the following properties:

(i)

$M^{t}$ is independent of $t$ , and is the result of adding a 2-handle to $T^{*}{\mathbin{\cal S}}^{3}$ . There is an isomorphism $H^{2}(M^{t},{\mathbin{\mathbb{R}}})\cong{\mathbin{\mathbb{R}}}$ identifying $[\omega^{t}]$ with $t$ . Thus $(M^{t},\omega^{t})$ is an exact symplectic manifold if and only if $t=0$ .
(ii)

For $t>0$ there is a compact, embedded Lagrangian $L^{t}$ in $(M^{t},\omega^{t})$ diffeomorphic to ${\mathbin{\cal S}}^{3}$ , depending smoothly on $t$ , with $0\neq[L^{t}]\in H_{3}(M^{t};{\mathbin{\mathbb{Z}}})\cong{\mathbin{\mathbb{Z}}}$ .
(iii)

There are no Lagrangian ${\mathbin{\cal S}}^{3}$ ’s in $(M^{0},\omega^{0})$ , and in fact, no compact, exact, embedded Lagrangians in $(M^{0},\omega^{0})$ at all.
(iv)

As in [25, Rem. 3.7], $L^{0}=\lim_{t\rightarrow 0}L^{t}$ is a singular Lagrangian in $M^{0}$ , which topologically looks like an ${\mathbin{\cal S}}^{3}$ with an ${\mathbin{\cal S}}^{1}$ collapsed to a point $p$ , so that topologically $L^{0}$ is modelled on a $T^{2}$ -cone near $p$ .

All this suggests that $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ is empty for $t=0$ , and nonempty for $t>0$ . This counts as pathological behaviour, discontinuous in $t$ , since the $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ for small $t>0$ are not deformations of $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm nonsing}$ in a meaningful sense. Intuitively, one would expect objects to disappear under small deformations owing to obstructions, so that $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ for $t>0$ should be smaller than $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm nonsing}$ .

It seems plausible that we can choose the $L^{t}$ up to Hamiltonian isotopy so that $L^{0}$ has one singular point $p$ locally modelled on $C$ in (2.4), and $L^{t}$ for $t>0$ is locally modelled near $p$ on $L_{1}^{A(t)}$ in (2.5), where $A(t)\rightarrow 0$ as $t\rightarrow 0$ . If so, $L^{0}$ may give an object in $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm sing}$ , and the derived categories $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm sing}$ may depend continuously on $t\in[0,\epsilon)$ . So in this example, $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ may be better behaved than $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ under deformations of $(M,\omega)$ .