I read Tom Braden's paper on mixed category for toric varieties. At least one idea is useful, namely where does the extra grading come from.

The idea is that, given a vector space with a unipotent operator, then you can split the vector space into Jordan blocks. Each Jordan block has a Z-indexed-filtration, with the index well-defined upto a global shift.

Suppose you have T^*P^n, with the stratification given by P^0, P^1, ... , like in category O. Then we want to equip the wrapped Fukaya category's cocore's endomorphism with extra grading.

Combinatorially, we can try to match the $End(T)$ algebra with the combinatorially defined $A(V)$ algebra. $A(V)$ algebra has extra grading. You can say that this is like $T$-equivariant cohomology, where the generator has cohomologically grading $2$. Then, we forget the cohomological grading and only keep the extra grading. Then the question is, why the endomorphism of $T$ brane is the same as $A(V)$. You can say that Gammage-Mcbreen-Webster solved the problem, but not quite. Local system is naturally identified with coherent sheaves module over torus; Only unipotent local system is identified with module near the identity of the torus; and only unipotent system with an explicit Frobenius action allows you to have a $\Z$-index filtration. So, the Fukaya category needs a graded lift.

There should be a version of the A-side, whose endomorphism of co-core is isomorphic to $A(V)$ as an ungraded algebra, then we can choose some graded lift. Well, the crossing over (between strands of different color) should have grading $1$, dot having grading $2$. That's the grading convention of $A(V)$.

Now, what's the relation between

• Frobenius on a variety $X$ defined over $\F_q$. we get, mixed constructible sheaf.
• algebraic Lagrangian skeleton in a algebraic symplectic variety, over $\F_q$. we should get mixed microlocal sheaf.
• holomorphic Lagrangian skeleton in a holomorphic symplectic variety, over $\C$. we should get mixed microlocal sheaf.
• Ask Saito, what's the analog in $\C$ for the mixed constructible sheaf?

In principle, all these question should be answered by these Koszul duality people. Koszul duality is about, two holomorphic skeletons, and a duality correspondence that sends cocore to core. It has been proven in the realization of mixed DQ module; but not yet in the relatization of Fukaya category.

First of all, Reeb chord, is not like Morse critical point. So, we don't necessarily have a canonical grading on it. It depends on a choice of holomorphic symplectic structure. There is some Conley-Zehnder index that I don't quite know. Maybe only $\Z/2$-grading is canonical. But that is Morse grading.

Let's dream a bit. Consider the 3d mirror between $T^*\P^2$ and $A_2$ surface, the smoothing of $\Z^2/\mu_3$ (we say $A_1$ surface is the smooth $T^\P^1$). We have a