1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. THE GEOMETRY OF THE MOMENT MAP 307

which has the polytope P as its moment polytope; if P has d faces of codimension
1, one constructs the toric manifold M as a symplectic quotient of a vector space
V ~Cd by the linear action of a torus T' ~ U(l)d-n. The torus T ~ U(l)n acting

on Mis then obtained by reduction in stages, as the quotient of U(l)d by T'. (See

[15], Chapter 1.)

Example 2.11. The moment polytope for the action of U(lr on cpn is the n-

simplex. This action descends from the action of U(l)n+l on cn+^1 , using reduction

in stages: recall from Example 1.11 that we constructed cpn as the symplectic

quotient of cn+i by the diagonal U(l) action.