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 with a unique intersection point , and there is a Darboux chart around modelled on , such that inside the chart the local setup is
Let be a 1-parameter family of Lagrangian connected sums with neck length , which all agree with except in a compact subset in . Inside , we take the ansatz
where the curve can be chosen so that is almost calibrated and agrees with outside . Clearly, are related by scaling inside . The Hamiltonian vector field along , which is really a section of , agrees with inside and is zero outside.3838 38 This is consistent because near the boundary of , the position vector is a tangent vector of , so vanishes in the quotient . The corresponding Hamiltonian function is times a smooth function of one variable ; a small caveat is that converges to two generally different constants along and . Thus it takes finite distance in the Solomon metric to reach the limit , and the Solomon functional remains finite, but the topology changes from to .
Remark 2.13.
The Hamiltonian functions along can be extended to global functions on , with
But in the limit, the second derivatives fail to be continuous at the origin, and indeed 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 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 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 class are expected to flow to the same limit, they are supposedly connected to each other through a continuous family of unobstructed Lagrangians.