• 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.