LECTURE 4
The Duistermaat-Heckman Theorem and Applications to
the Cohomology of Symplectic Quotients
Let M be a symplectic manifold equipped with the Hamiltonian action of agroup G. The element w E Db(M) defined by
w(X) = w + (μ,X)
satisfies Dw = 0 and thus defines an element [w] E Hb ( M).
Theorem 4.1. (Duistermaat-Heckman theorem, version II) Suppose M is
a symplectic manifold of dimension 2n equipped with the Hamiltonian action of a
torus T. Then for generic XE t, in the notation of Theorem 3.10, we have
{ (iw)n eiμ(m)(X) = "'""' eiμ(F)(X) { ~-
j M n! ~ J F eF(X)
FEF
Proof: Apply the abelian localization theorem (Theorem 3.10) to the class exp iw E
H(;(M). (By Example J..5, for each component F of the fixed point set of T, the
value of μ(F) is a constant.) D
4.1. Stationary phase approximation
An alternative approach to this version of the Duistermaat-Heckman theorem ( "ex-
actness of the stationary phase approximation") is sketched as follows. Assume for
simplicity that T = U(l) and that the components F of the fixed point set are
isolated points. By the equivariant version of the Darboux-Weinstein theorem [38],
we may assume the existence of Darboux coordinates (x 1 , y 1 , ... , Xn, Yn) on a coor-
dinate patch UF about F, for which μ(x1,Y1, ... ,xn,Yn) = μ(F)-2:.::j ~(xJ+yJ).
Thus the oscillatory integral over U F tends (if we may replace U F by IR^2 n) to
(4.1) 1 eiweiμ(m)X = { indx1dY1 ... dxndyneiμ(F)Xe-iL,JmJ(x]+y])X/2.
mEM ~h
The integral over IR^2 n is given by a standard Gaussian integral:
e'weiμX = d~f SF(X)
1.. (27r)neiμ(F)X
JR2n (IJj mj )Xn
315