1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

300 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

t E t* of the moment map μ; the space Mt is a symplectic orbifold for any regular

value t ofμ.

Example 1.11. Let U(l) act diagonally on cn+l equipped with the standard sym-

plectic structure. The moment map for this action is

1 n

μ(zo, ···,Zn)= -2 2..: lz11

2

,

j=O

so the symplectic quotient μ-^1 (-~)/U(l) is complex projective space

s2n+1 /U(l) ~ <CPn.

More generally we may consider the reduced space M>-. = μ-^1 (0>-.)/G when

0 >-. is the orbit in g* through .A E g* ( coadjoint orbit) All such orbits may be

parametrized by .A E t~ where t~ is a chosen positive Weyl chamber in t*.

Example 1.12. The shifting trick. Let .A E g*. We define a symplectic structure

w^0 >- on the coadjoint orbit 0 >-. (in terms of the vector fields X#, y# generated

by the action of X, YE g) by wf >-(X# >-., y# >-.) = -.A([X, Y]) at the point .A E 0>-.

(and everywhere else on the orbit by equivariance). The moment map for the action

of G on 0 >-. with respect to this symplectic structure is the inclusion of 0 >-. in g*.

(The symplectic structure on the orbit was found by Kirillov and Kostant; see for

instance Section 7.5 of [6].)

Define a symplectic structure D on M x 0 >-. by

D=wM -w^0 >-.

Then for the moment map with respect to the induced action of G on M x 0 >-. we

have

M>-. ~ (M x 0>-.)o.

Corollary 1.13. Combining Example 1.9 with Proposition 1.6 (b) we see that for

any linear action of a group G on cpn-^1 (i.e. an action factoring through a

representation G---+ U(n)) the moment map factors as

μ=nofl

where fl : cpn-l ---+ u( n) was given in ( 1. 1) and 7f : u( n) ---+ g* is the projection

map.

In particular one often demands for a projective manifold M (i.e. a compact
complex manifold with an embedding into <e,pn-l) that the action of G extends to
a linear action on <e,pn-l. Thus moment maps for such linear actions are given by

(1.1) composed with n and with the embedding of M into cpn-^1.

1.3. The normal form theorem

There is a neighbourhood of μ-^1 (0) on which the symplectic form is given in a

standard way related to the symplectic form w 0 on Mre d (see for example sections
39-41 of [21]).

Proposition 1.14. (Normal form theorem) Assume 0 is a regular value ofμ

(so that μ-^1 (0) is a smooth manifold and G acts on μ-^1 (0) with finite stabilizers).

Then there is a neighbourhood 0 ~ μ-^1 (0) x {z E g, lzl :::; h} ~ μ-^1 (0) x g of