====== Torus acting on a Toric Variety ======


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 $X$. 


**Theorem **
$$ Loc(X)^{Loc(T)} \cong \mu sh(\wt \Delta_X) $$
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. 



===== Example =====
For example, $X=\C$, $z=x+iy$, $S^1$ acts on $X$ by rotation $x \d_y - y \d_x$. The actual space is $T^*X$, with base coordinate $z$, fiber coordinate $\xi = p_x - i p_y$ as fiber coordinate. The symplectic form is $\Omega = d z \wedge d\xi = dx \wedge dp_x + d y \wedge d p_y + i(...)$, and $\omega = Re(\Omega)$. This $\omega$ is not Kahler for this complex structure, don't worry. We have $\C^*$ rotate the base and fiber in the opposite way, $t(z,\xi) = (tz, t^{-1} \xi)$, and we have Hamiltonian function $H_\C = z \xi = (x p_x + y p_y) - i (x p_y - y p_x)$. The real Hamiltonian function is $H = x p_y - y p_x = -Im H_\C$. If we do honest symplectic reduction at moment map value $0$, we should do $H^{-1}(0)/S^1$. 

How should we understand this quotient space? Suppose the coordinate $z, \xi$ lives in $I$-complex structure. We now need to go to the $J$-complex structure, which makes $\omega_J = dx \wedge dp_x + dy \wedge dp_y$ kahler, i.e, $u = x + ip_x, v = y+ip_y$ now become $J$-hol coordinates, then $H = Im( \bar u v)$. Now, how to extend this $S^1$-action to a $\C^*$-action, well the generator is $u d_v - v \d_u$. We need to introduce new $J$-hol coordinate $u+iv$ and $u-iv$. We have 
$$ A=u+iv = (x+iy) + i (p_x + i p_y) = z + i \bar \xi, \quad B=u - i v = \bar z + i \xi $$
This change is a unitary change of variable, hence preserves the standard hermitian form (up to a factor of 2). Thus, the Hamiltonian function is $H = |A|^2 - |B|^2$. 
Under this $\C^*$ action, such that $t (u+iv, u-iv) = (t (u+iv), t^{-1}(u-iv))$, we have the invariant function is $AB=(u+iv)(u-iv) = u^2+v^2$. Here, we can use symplectic reduction for Kahler manifold equal to GIT quotient, to get $\C^2_{A,B} /_0 \C^* = \C$. This is one of the funny moment, where $\C^*$ acts on $\C^2$ by weight $(1,1)$, and you know that the perturbed quotient would give you honest $\C$, but the 'singular' quotient would give you the seemingly smooth $\C$ as well. 

To summarize, we are working with $\omega_J$, and we use $J$ complex structure to make it Kahler, to do usual Fukaya category. A better thing to do is to work with $\Omega_I$, the holomorphic symplectic form, and consider general $\omega_\theta = Re(e^{i\theta} \Omega)$, and do Fukaya category with that. 

So, you see, it is funny. This $S^1$-action, is a $I$-complex Hamiltonian action, with $\Omega_I$ Hamiltonian $z \xi$. 

===== 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).