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--- projects:torus-action-on-fukaya [2022/10/28 07:27] – created pzhou
+++ projects:torus-action-on-fukaya [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 Let $X$ be an complex $n$ dimension smooth toric variety, and let $T = U(1)^n$ the compact torus acting on $X$. Assume $X$ is equipped with a $T$-invariant symplectic form $\omega$, and $\Delta_X$ is the moment polytope of $X$.
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 $$ Loc(X)^{Loc(T)} \cong \mu sh(\wt \Delta_X) $$
 where $\wt \Delta_X \to \Delta_X$ is certain FLTZ Lagrangian skeleton standing over $\Delta$, with $n-k$ dimension torus fiber over the $k$-dimensional boundary strata.
-** Remark**
-  * The grading will be a bit funny, let's say we work with $\Z/2$-graded categories.
-  * The symplectic thickening needs some thought. For example, if we have a gluing of toric varieties $X_1 \cup X_2$ along some other toric varities $X_{12}$, where $X_{2}$ is in the cotangent direction of $X_1$, then, after quotient, we should get $X_1/T$ and $X_2/T$ on the 'same side' of $X_{12}/T$.
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 ===== How to prove this =====
+** Remark**
+  * The grading will be a bit funny, let's say we work with $\Z/2$-graded categories.
+  * The symplectic thickening needs some thought. For example, if we have a gluing of toric varieties $X_1 \cup X_2$ along some other toric varities $X_{12}$, where $X_{2}$ is in the cotangent direction of $X_1$, then, after quotient, we should get $X_1/T$ and $X_2/T$ on the 'same side' of $X_{12}/T$.
+Locally, this is a product, no worries.
+===== Generalization =====
+This is a special case of hypertoric variety, where we take $T^*X$. In general, we can have more complicated ones. No, I don't want anything infinite type, like infinite chain of $\P^1$.
+Consider $\C^2 \RM \{xy+1=0\}$. Here, $x,y$ are the $J$-hol coordinates, like the $A,B$ above before, since the quotient map is map to $xy$, and the $\C^*$-action is scale $x,y$ in opposite way.
+If we do it brutal force, define $xy+1=e^u$, then for $u \in 2 \pi i \Z$, we have singular fiber.
+I guess the Hamiltonian function is still $H = |x|^2 - |y|^2$.
+Somehow, $\Omega_J = dx \wedge dy / (xy+1) = e^{-u} dx dy $. Can we talk about real and imaginary part of it? But we don't want these.
+I want to have $I$ complex coordinate. Well, the space will not be affine.
+I probably should consider additive $T^*\P^1$ first, Well, $T^*\P^1$ is the resolution of $A_1$ singularity, where we consider $\C^2/\Z_2$. We can keep going, consider $\C^2/\Z_n$, and take certain colimit, by open inclusion.
+This is an infinite Weinstein handle attachment process. Or, do we attach on both ends? We could. Then, there is a $\Z$ translation action on the limit (but not on the finite truncation).