Clarification on the ”neck stretching” applied to the base space of a Lefschetz fibration

Question

I’m asking this question because I’d like to understand better the neck-stretching argument in symplectic geometry and what kind of conclusions one might get out of it in my setting.

Assume that I’ve a Lefschetz fibration $E$ over $M$, where $M$ is an exact symplectic manifold (maybe with ends). Let $C$ be a contact type codimension $1$ submanifold of $M$. As far as I understood it, if I stretch the neck around $C$ I create a family of manifolds $M_t$, $t\in \Bbb R^+$ (with $M_0=M$) all symplectomorphic to $M$, “representing the stretching of the neck” around $C$ at time $t \geq 0$.

According to the thesis of Evans [E], this process tend to a limit $M_{\infty}$ which is obtained from $M$ by cutting along $C$ and gluing the positive/negative part of the symplectisation of $C$ (in the thesis it’s denoted as $\Bbb S_{\pm}(C)$. I do believe that similarly this process should give rise to a family of Lefschetz fibrations $E_t$ over $M_t$ and a limiting fibration $E_{\infty}$ (I believe that we attach some trivial fibration over the ends we are introducing).

I’m interested in understanding the moduli space of pseudo-holomorphic sections of such fibration. Let us assume that I’ve chosen a generic compatible a.c. Structure $J$ and I remember that I don’t have bubbling due to my exactness assumption. Hence my stretching procedure gives rise to a family of manifolds $\mathcal{M}_t$. What can I say about this family of moduli spaces? How do I treat the limiting case $\mathcal{M}_{\infty}$? Is it somehow a boundary piece for the family of spaces parametrized by the positive reals?

I think I can come up with some naive reasoning supporting my claim, but since I’m trying to push something to the limit I’m expecting a lot of technicalities involved and I can’t appreciate them yet.

Since I’m rather new to the field, can someone points me if there are any obvious obstruction/difficulties in what I’m trying to understand and if there are any literature that Somehow sheds some light on what I’m trying to understand?

Thanks in advance for any comments!

References

[E] Jonathan David Evans - Symplectic topology of some Stein and rational surfaces

Answer 1

For a start, the base of your Lefschetz fibration had better be a Riemann surface, or else it won't have any pseudoholomorphic sections for generic J (see for example Kruglikov's paper https://link.springer.com/article/10.1007/s00229-002-0352-2). If the base is a Riemann surface then stretching the neck is just a special deformation of the conformal structure on the Riemann surface. Paul Seidel set up the theory behind counting pseudoholomorphic sections of Lefschetz fibrations over Riemann surfaces (with Lagrangian boundary conditions) in his early papers:

https://arxiv.org/abs/math/0105186 (Section 2)

https://arxiv.org/abs/math/0309012 (Section 3)

and his book "Fukaya categories and Picard-Lefschetz theory" (see, e.g. chapter 9). In his work, he allows arbitrary deformations of the conformal structure (not just neck-stretches). As you anticipate, as the surface degenerates into a Deligne-Mumford style bubble-tree, the sections degenerate into sections over the components. This is what lets you set up the $A_\infty$-structure on the Fukaya category, for example (where the Lefschetz fibration is just a product (no singular fibres)).

If you really want an example of a paper where someone calls this "neck-stretching", you can see it in Tonkonog's paper

https://arxiv.org/abs/1504.01621

where he needs some specific family of Riemann surfaces to act as domains for his pseudoholomorphic curves (i.e. you don't allow all deformations of the conformal structure); this kind of trick is often used to ensure that particular kinds of degeneration appear with virtual dimension zero (if you allow all deformations then the things you want to count might have positive virtual dimension).