1. The problem statement, all variables and given/known data Given a 2-torus action on C^3 can be defined by (a,b).(x,y,z)= (abx, a^-1by, bz). What is the moment map? 2. Relevant equations 3. The attempt at a solution Here the Lie algebra is isomorphic to the dual Lie algebra which is dual to R^2. I was trying to compute the moment map directly. The vector fields that generate the action in polar coordinates on C^3 are (I think): 1) partial w.r.t theta_1 - partial w.r.t theta_2 2) partial wrt theta_1 + partial wrt theta_2 + partial wrt theta_3 3) partial wrt theta_3 Why do I think those: I'm first thinking of b=1 & letting a vary, then for the other one I do the reverse, a=1 & b varies. Then do calculation of -(interior derivative of "1)")omega = ... (some work I'm leaving out for now) = d( |x|^2 /2 - |y|^2 /2 ) Similarly for "2)" & "3)". So end of day I think the moment map is: u(x, y, z) = (|x|^2/2 -|y|^2/2 , |x|^2/2 +|y|^2/2 +|z|^2/2, |z|^2/2 ) I also think I've done something wrong in this attempt.