Peng Zhou

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going to revise the VGIT paper.

Here, the base is one dimensional, fiber is easy. We can take the easy way out, just prove enough for this case. Or, we can do full proof.

I don't want to be super general, but just for this case. I want to consider skeleton on the total space, and skeleton on a sub-level set.

let's just glue category, that is pretty cool, and useful.

So, we have a constructible sheaf of categories. I think, we can define a sheaf of stable categories. We don't have the notion a microlocal stalk, but we do have the notion of equivalences. In that sense, it is not very satisfactory.

I need to say, non-characteristic deformation lemma. what is 2.7.2?

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well, two months passed.

what did I learn today? with Alexei's talk yesterday and the discussion today.

what I

fk, a month has passed.

what do I want? I want raising and lowering operator, which is adding and removing strand operator, which is a special case of gluing an extra guy and put some extra strands operator.

How does sl2 work? We know that, by KLRW algebra, the planar version, modulo some stop, there is only one way to realize the categorification.

I bet, we can do a purely downstairs theory, even define it.

So, how do we define it? Suppose, we say that hom betewen T-branes follows KLRW algebra, then what?

all the homs are in degree zero?

1. after discussion with M yesterday, I realized I need more terms in the differential, ok, not bad. I should write up some examples, for other people to get it.

2. I don't know how it is related with representations of $gl(1|1)$. I think it is about Alexander polynomial. (is it about oriented knot?)

so you found a secret rule to define the differential, ok, good for you!

how to prove that it works? they are guesses, though very solid ones.

1. you need to define a functor;

2. you need to prove an excision lemma, things are only dependent on the boundary, which reduces this question to the many stop case.

ok, why it is about $gl(1|1)$? Vera told me, we have $E,F, H, C$.

The removing strand operator is not that simple: taking intersections, and putting in the object. There must be some interesting differentials correcting it.

Let $L^k$ be a k-tuple of Lagrangians in $\Sym^k(\Sigma)$, avoiding a stop. Let $E$ be the raising operator, i.e., $F$ the lowering operator. Let $F$ be adding a brane, by adding a T-brane, and $E$ be $Hom(T, -)$. (note the change of notation).

Then, $F$ is easy to achieve, thanks to the stop, we can just add a brane there. But, $E$, its right-adjoint, is a bit difficult. The adjoint condition basically involves solving an equation. What we want, is a concrete, purely Fukaya category like functor.

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The new exciting thing today is, the raising and lowering operator.

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Now, what is the expectation? We should use the KLRW algebra as the benchmark to tell me what grading on the endo of the T-brane I should get. Basically, dot has positive grading 2, crossing with a puncture has grading 1.

First question, you claim that, you have an $S^1$-family of symplectic form, show me. Consider $\C^2 / \mu_n$, with weight $(1,-1)$, $n=2$. We consider the coordinate ring $$ \C[x,y]^{\mu_2} = \C[xy, x^2, y^2] $$ when you blow-up, you put in the ratio coordinate $u = x/y$ and $v = y/x$, you find that this happens to be invariant since $n=2$. So, you have two patches, one with coordinate $(x/y, y^2)$ call it $U_y$ (it does not mean $y \neq 0$, it just means when $x,y$ both goes to 0, $x$ goes to zero faster, the other with coord $(y/x, x^2)$ call it $U_x$. Is this the total space of $O(-2)$? yes (tangent bundle to fano has section, since we have many automorphism)

So, now, what is the hol'c symplectic form? We can use the old one, like $$ \Omega = dx \wedge dy = (1/2) d(x/y) \wedge d(y^2) = (1/2) d x^2 \wedge d(y/x). $$ It is the one that makes sense in local coordinate.

Suppose we do $\C^2 / \mu_3$, weight $(1,-1)$, then the invariants are $x^3, y^3, xy$. How do we blow-up? Well, we can consider more invariant functions, like $x^2/y, x/y^2, y^2/x, y/x^2$. So, we have 3 patche, with coordinate

• $(y/x^2, x^3)$.
• $(y^2/x, x^2/y)$
• $(y^3, x/y^2)$

I don't know how I get it, I guessed it, and it worked.

So, we have local coordinates, and we have holomorphic symplectic form. (why it is non-degenerate? well, just check it locally. )

We define $\omega_\theta = Re(e^{i\theta} \Omega)$.

What does extra grading mean? Well, you can have if you have a Kahler manifold $M$, you can have $H^*(M)$ equipped with an extra grading $p-q$, Hodge grading. Is it possible to get this from the $S^1$-family? In the case of $\P^n$, all the weight grading are zero, because you only have $(i,i)$ class. That's not the case for elliptic curve, where you have $(1,0)$ and $(0,1)$.