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--- blog:2023-02-26 [2023/02/27 07:52] – pzhou
+++ blog:2023-02-26 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 32: / Line 32: @@
 $$ Fuk((\C^*)^2, y_1 + y_2 + y_1 y_2) $$
 It is definitely not generated by the critical point. What is the fiber at $y_1 + y_2 + y_1 y_2 = 0$? Well, the fiber is then $\C^* \RM \{-1\}$. Generic fiber is a $4$ punctured $\P^1$, and special fiber is a $3$ punctured $\P^1$. This is also visible from the change of the Newton polytope for the defining hypersurface, over the value 0, one lose the constant term.
+In these cases, it makes sense to ask, on the B-model side, what is the endomorphism ring of the structure sheaf. That is the case if we have Coh as the B-side. Not so much when we have MF. So that we know, what is the weight of each endomorphism.
+===== GIT quotient of MF ? =====
+Suppose we have some basic MF on $\C^N$, like $W = x_1\cdots x_N$. And suppose we have some $\C^*$-action, say, given by weights $a_1, \cdots, a_N$, such that $\sum_i a_i = 0$, so $W$ is invariant. We can ask: what's going on here?
+Say, we have $(1,1,-2)$ weights. What is MF? It is some two-periodic chain complex, living on $W=0$.
+Is there a know-it-all sheaf, like structure sheaf on MF category? No, unfortunately no. Instead, we have three basic MFs.
+For the non-equivariant MF, there are three basic ones. And they have $(\C^*)^3$ equivariant lifts. Such lifts collapse to $\C^*$-lift for each $\C^* \to (\C^*)^3$. We have various weights. If $x,y$ has weight $1$ and $z$ has weight $-2$, then the three basic ones has different weights $(x,yz), (z,xy), (y, zx)$. The $(z,xy)$ one is between $O(k)$ and $O(k+2)$. Somehow, they don't fit from a window, and cannot be used.
+What if we choose some different weights? Like $(2,3,-5)$? Then, the window size is $5$. We cannot use $(z,xy)$, since it has size $5$.
+I think we should take MF first, then take GIT quotient.
+Consider the example of $(1,1,-1,-1)$, say with variable $x,y,z,w$. Window size is 2. Allowed factorization $(x, ..), (y, ...), \cdots$ 4 of them, and $(xz, yw), (xw, yz)$. two of them. All six makes geometric sense on the A-side.
+No, we should not use window so early. We are talking about GIT quotient. We should talk about polarization, unstable orbits. So what are those? Is polarization still about a line bundle?
+===== equivariant MF =====
+Following Segal.
+What's hom between MF? It is the dg hom between curved 2-periodic chain complexes, the beautiful thing is that, the curvature cancels out, and the result is an ordinary 2-periodic chain complex.
+Then, you take global section to get usual hom. Of course, sheaf hom is better.
+But, how to deal with removing unstable loci? What's the approach of Segal? Well, he took GIT quotient first, without worrying about $W$. Then, the usual window works. Lemma 3.5 there.