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--- blog:2023-02-26 [2023/02/27 06:12] – pzhou
+++ blog:2023-02-26 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 13: / Line 13: @@
 We can say $Fuk(Y, W_Y)$ is mirror to $Coh(X, W_X)$. We have $T$ acts on $X$ preserving $W_Y$, and $Y$ maps to $T^\vee$ (how does this map interact with $W_Y$ ?).
+In the following, what I wanted should be true for general $X, Y$, not necessarily toric (they can come from toric guys as subvariety), only need some torusaction, and torus invariant.
+What's the trouble? Well, we know very well that in the B-side, we need to choose some point in the G-equivairaint ample cone, and assuming affine, some character of $T$. I just need some simple enough reason to say that, this choice, a cocharacter of $T^\vee$, means taking certain tropical limit in that fibration.
+Why is that?
+===== Non CY case, again =====
+If B-side has inequivalent quotient, how does A-side say?
+** Ex 1 ** Say B side, we have weight $(1,1)$ for $\C^*$ acting on $\C^2$. On the A-side, we have $W = y + t/y$ over space $\C^*_t$.
+  * On the B-side, the quotient is $\P^1$ or empty set
+  * On the A-side, the mutual critical fiber for $(y_1+y_2, y_1y_2): (\C^*)^2 \to \C \times \C^*$ has critical point at the diagonal $y_1=y_2=y$, then the image of the critical loci is $W^2 = 4t$.
+** Ex 2** Another example, where we have $\C^*$ acts on $\C^3$ by weight $(1,1,-1)$, the mirror superpotential have $t = y_1 y_2 / y_3$ with $W = y_1 + y_2 + y_3$, and we have $W = y_1 + y_2 + y_1 y_2 / t$. So, we have critical loci, which is at
+$$ 1 + y_2/t = 0, \quad 1 + y_1/t = 0 $$
+so $y_1 = y_2 = -t$ is the critical loci. Pick any $t$, say $t=1$. What is
+$$ Fuk((\C^*)^2, y_1 + y_2 + y_1 y_2) $$
+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.