• Paper revision
• Nagao and Nakajima. Transition of Conifold, and DT transformation.

Mirror symmetry intertwines equivalences transformation between A-side and B-side.

In the case of $\C^N / \C^*$, we know how the B-side works (via toric window), and how the mirror symmetry work, via (CCC), so here is how the A-side will work.

“This work is complete and compelling” and “also with a very useful sharpening of the singular support estimate”. OK!

Now, here are the problems

Where is the key estimate $$ SS(\pi_* Sh(\La_B)) \In \pi_*(SS(Sh(\La_B)) $$ established?

A similar estimate for sheaves valued in stable category is well-known (hmm, my stuff is also stable, right? closed under cones etc). So, is my notion of singular support different from the standard one?

Hmm, I justed bluffed that this is obvious, but now I am caught here. But, why is it not obvious? Naively, the singular support of a sheaf of anything is defined by the nearby cycle functor, no?

OK, I see. The referee is complaining I am using a seemingly different notion of singular support for categories than Kashiwara-Schapira. Indeed, when the space is stratified, and we have a priori bound, then maybe we can use the old notion.

OK, indeed, I will say singular support for stable categories as is, and only remark that in case of constructible sheaves, this can be checked without much effort.

Not a big deal :)

Aha, classically generate means the smallest stable subcategories that contains these object.

Good question, why do they generate, well because back in $\La_{\C^N}$, they compactly generate the large category.

As David Nadler says “the only nice construction in category is universal construction'.

ok,this is already on p22.

Not a big deal, just need to add: because in this grid stratified case, singular support can be estimated easily.

1. I really need to be precise! ahh. being complained. so, let me be more precise. Constructible sheaf with stratification $S$ is related to Lagrangians with singular support $\Lambda_S$.

2. yes, Segal introduced this notion.

3. better reference habit, give precise reference. Yes, we understand you are lazy, but please don't be lazy.

4. give reference to the later introduced FLTZ skeleton.

5. indeed, the notion of partially wrapped CCC is new. let me think how to recover that. I think this is best done by this: starting from the equivariant CCC, then on the B-side, we de-equivariantize on a subgroup, meaning we forget the equivariant structure.

...

6. yes, forward reference about symplectic reduction should be given.

7.