Again, considering the VGIT and LG problem. Read a bit BFK in the morning. The graded MF is really something.

Another thing that I realized is, at least for HMS of toric GIT, the choice of a cocharacter in $T_B$ corresponds to a 1PS in $T_A$. One probably should consider compactifying the base.

But I don't want to compactify, because I don't know what is the limit of that holomorphic function $W_t$.

When in doubt, do graph. I think that is a good approach. For example, consider the function on $(\C^*)^2$ $f(x,y) = x(1 + y + ty^2)$. As $t \to 0$, we know the term $t y^2$ will be important eventually when $y$ is large enough. So, we can include $t$ as part of the variables, and we have some terms in $\Z^3$. We are going to look at those 1PS, where $t$ variable has speed 1.

a la Webster.

First recall how it was done on the B-side space. Let $G=\C^*$ acts on my space $X$ on $B$-side. Let $L$ be an $G$ equivariant ample line bundle. We consider the projective ring of $\oplus_n \Gamma(L^n)$, then taking $G$-invariant subring.

Next, consider what happens on the categorical level. Category is nothing but relations. You don't say points and value. You only say hom among objects.

now, suppose you have a sheaf, support in the unstable loci, what happens? For example, $\C^*$ acts on $\C^2$ with weight $(1,-1)$, and you have a point $p = (0, 1)$, and you choose $+$ as the chamber. It means, you only consider the graded ring with function $x^a y^b$ of $a > b$. In particular, you have function $x, x^2,\cdots$, but not function $y, y^2$. And this poor guy is always zero in your eyes.

What do we do? I want to create module. So, is the following functor going to give me module? I am going to take the ring $\oplus_{n \geq 0} Hom(O, L^n)^{\C^*}$. I know $Hom(O, O)$ will have function like $x^a y^a$ as eq-degree $0$. $L = O \otimes \chi_1$. So it is like $x^a y^b \otimes \chi_1$ will have weight $-(a-b)+1$ (for some convention of sign), and we ask this to be $0$. So, $a-b=1$.

Given a sheaf $F$, say $G$-equivariant, we can consider the various $\wt F = \oplus_{n \geq 0} F \otimes L^n$. Say, everything is affine, so I will take its global section, and G-invariant. I should get a graded module, also denoted as $\wt F$. If I have a $\alpha \in O(3) = Hom(O, L^3)$, and $f \in F(4) = Hom(O, F \otimes L^4)$, I should be able to increase the twist to get a map $F \otimes L^4 \to F \otimes L^7$. That's it. No need to take global section on the second guy.

Notice that, $F, L$ they are already in the equivariant category, so in the stacky quotient if you wish. It is not working in the original category anymore. So, one cannot talk about points. The only thing meaningful is $G$-orbit, of various kinds.

Now, let's consider the categorical version. So, we have the $G$-equivariant category, $C^G$. Tensoring by certain equivariant line bundle is an automorphism (invertible endomorphism). Call this functor $S = - \otimes L$. It is intresting to consider $Hom(Id, S^k)$ the collection of natural transformations, which can compose. That's the projective 'center'? Just like $Hom(Id, Id) = HH^*(C)$ is the center.

We need to find the bad guys in $C^G$, those objects that is bad in the eye of $S$. That means, for any $\alpha in Hom(O, S^k)$ for $k>0$, the morphism $\alpha: F \to S^k(F)$ is zero.

For example, let $F = \C[x,y] / (x)$. This should not survive, since for the + quotient, only sheaf on $x \neq 0$ will survive. Let me pretend that $x$ is a morphism of degree $-1$. So, we have that quotient $$ F = [O(-1) \xto{x} O(0)] $$ So, for $x \in Hom(O, S)$, we have $x_F: F \to F(1)$ and that is zero. ok, so $F$ is dead.

We may define the subcategory to be kill as, things killed by all the positive degree morphism in $End(S)_{>0} = \oplus_{n>0} Hom(O, S^n)$.

Finally, let's say what this means on the equivariant A-side. $S$ is the shift of the equivariant positiion to the right. so indeed, on the constructible sheaf side, it worked. Now, how to get a morphism? Well, there is only one morphism in the base, between two lines, and there might be more in the fiber.

In general, for more general GIT quotient by $(\C^*)^n$, what's the story?

OK, let's consider this. On the B-side, we have $\C^N$. The structure sheaf is mirror to the totally positive real line.(when we do the actual mapping, need to backward wrap a tiny little bit.)

What's its shadow on the base $((\C^*)^k)^\vee$? Well, it is the positive real line. Is it really away from the discrimiant locus?

Consider hte equation for $1+x+y+t xy = 0$, where $t$ is the parameter, when is the equation singular? it is at $t=1$. darn it. why it has to pass through the critical point?

Consider the case of $(1,1,-2)$, window size is 2.

OK, what we do? In the toric case, we pick a character on the B-side GIT quotient, hence a 1PS on the A-side base, in particular, a loop, an element in HH. or projective HH. We do the shift, look for the morphism in the wrapped category