Differences

This shows you the differences between two versions of the page.

--- blog:2023-02-25 [2023/02/26 15:51] – pzhou
+++ blog:2023-02-25 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 29: / Line 29: @@
 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?
+===== Universal Window? ======
+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.
+===== B-side Unstable Loci =====
+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
+===== Non CY case? =====