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Exact isotopy class versus derived category class [048W]

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Exact isotopy class versus derived category class

The example below is closely related to the most symmetric case of the Lawlor necks (cf. section 2.3). It is also morally related to the Lawlor neck pinching singularity in the Joyce program [41, section 3.5].

Example 2.12.

Consider two almost calibrated Lagrangians L1,L2L^{1},L^{2} with a unique intersection point pp, and there is a Darboux chart around pp modelled on B1⊂ℂnB_{1}\subset\mathbb{C}^{n}, such that inside the chart L1,L2L^{1},L^{2} the local setup is

L1=ei​π/n​ℝn,L2=ℝn,ω=−12​∑d​zk∧d​z¯k,Ω≈∏d​zk.L^{1}=e^{i\pi/n}\mathbb{R}^{n},\quad L^{2}=\mathbb{R}^{n},\quad\omega=\frac{\sqrt{-1}}{2}\sum dz_{k}\wedge d\bar{z}_{k},\quad\Omega\approx\prod dz_{k}.

Let (Lt)0<t≪1(L_{t})_{0<t\ll 1} be a 1-parameter family of Lagrangian connected sums with neck length O⁡(t)O(t), which all agree with L1∪L2L_{1}\cup L_{2} except in a compact subset in B1B_{1}. Inside B1B_{1}, we take the ansatz

Lt={(t​γ​(s)​x1,…,t​γ​(s)​xn)|x12+…​xn2=1},L_{t}=\{(t\gamma(s)x_{1},\ldots,t\gamma(s)x_{n})|x_{1}^{2}+\ldots x_{n}^{2}=1\},

where the curve γ⁡(s):ℝ→ℂ\gamma(s):\mathbb{R}\to\mathbb{C} can be chosen so that LtL_{t} is almost calibrated and agrees with L1∪L2L_{1}\cup L_{2} outside B1/2B_{1/2}. Clearly, LtL_{t} are related by scaling inside B1B_{1}. The Hamiltonian vector field along LtL_{t}, which is really a section of (T​X/T​Lt)|Lt(TX/TL_{t})|_{L_{t}}, agrees with (γ⁡(s)​x1,…,γ⁡(s)​xn)(\gamma(s)x_{1},\ldots,\gamma(s)x_{n}) inside B1B_{1} and is zero outside.3838 38 This is consistent because near the boundary of B1B_{1}, the position vector is a tangent vector of LL, so vanishes in the quotient T​X/T​LtTX/TL_{t}. The corresponding Hamiltonian function hth_{t} is t2t^{2} times a smooth function of one variable ss; a small caveat is that hth_{t} converges to two generally different constants along L1L_{1} and L2L_{2}. Thus it takes finite distance in the Solomon metric to reach the limit t→0t\to 0, and the Solomon functional remains finite, but the topology changes from LtL_{t} to L1∪L2L^{1}\cup L^{2}.

Remark 2.13.

The Hamiltonian functions hth_{t} along LtL_{t} can be extended to global functions on XX, with

‖ht‖C0=O⁡(t2),‖d​ht‖C0=O⁡(t),‖∇2ht‖L∞≤C.\left\lVert h_{t}\right\rVert_{C^{0}}=O(t^{2}),\quad\left\lVert dh_{t}\right\rVert_{C^{0}}=O(t),\quad\left\lVert\nabla^{2}h_{t}\right\rVert_{L^{\infty}}\leq C.

But in the t→0t\to 0 limit, the second derivatives fail to be continuous at the origin, and indeed LtL_{t} changes topology in the limit.

In this example the essential failure is the breakdown of smoothness. In view of Joyce’s program, this suggests that the remedy is to allow for (Floer theoretically unobstructed) almost calibrated Lagrangians connected to each other not just by exact isotopies, but also surgeries such as Lagrangian connected sums. In these transitions the derived category class of the Lagrangian is unchanged, and the Thomas-Yau argument suggests the Db​F​u​k​(X)D^{b}Fuk(X) class is a natural framework to look for special Lagrangian representatives. The following fundamental question is thus relevant for the compatibility between the geometric and the categorical perspectives:

Question 2.

When are two almost calibrated unobstructed Lagrangian branes isomorphic in Db​F​u​k​(X)D^{b}Fuk(X) connected by exact isotopies with surgeries?

Remark 2.14.

The Joyce program suggests that running the LMCF would result in a sequence of exact isotopies and surgeries, to connect the initial Lagrangian to its infinite time limit, which one hopes to be the unique representative of the Harder-Narasimhan decomposition. In the almost calibrated case, there is no ‘collapsing zero object’ in this process for homological reasons, so the surgeries should be continuous in the geometric measure theory sense. Since any two such Lagrangians within the same Db​F​u​k​(X)D^{b}Fuk(X) class are expected to flow to the same limit, they are supposedly connected to each other through a continuous family of unobstructed Lagrangians.

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