Logarithmic convexity of push-forward measures

Abstract

Let M be a compact symplectic manifold with a Hamiltonian action o f a compact torus T, and moment map q5 : M --~ t*. The push-forward by q5 of the symplectic measure on M equals Lebesgue measure on q~(M) multiplied by an "interpolating function" f . These interpolating functions are important because if M is a smooth projective variety, and the symplectic form on M represents the first Chern class o f a T-equivariant ample line bundle L -~ M, then the interpolating function f describes the asymptotic behavior of the multiplicities o f the representation of T on H~ | ([He], [GS2]). One striking property of the moment map is that its image qS(M) is a convex polytope in t* ([A], [GS1]). The purpose of this paper is to show that if the symplectic form is K~ihler, then the interpolating function also possesses a certain convexity property. Recall that a continuous function h on a convex set C is called convex if