Kaehler structures on toric varieties

VICTOR GUILLEMIN 1.Let (X, ω) be a compact connected 2w-dimensional manifold, and let (1.1) τ: T n -+Όifί(X, ω) be an effective Hamiltonian action of the standard w-torus.Let φ: X -> R n be its moment map.The image, Δ, of φ is a convex polytope, called the moment polytope.Delzant showed in [5] that the triple (X, ω, τ) is determined up to isomorphism by this polytope, and also that X has an intrinsic T n -invariant complex structure which is compatible with ω and makes X into a toric variety.The purpose of this note is to show that is not only the symplectic geometry of X determined by Δ, but also, to a certain extent, the Kaehler geometry of X.By [5], Δ can be described by a set of inequalities of the form (1.2) (x,W;)>V ι=l,... ,</; the u.'s being primitive elements of the lattice, Z n , and d the number of (n -l)-dimensional faces of Δ.Let /,: R n -> R be the map and let Δ° be the interior of Δ.Then x e Δ° if and only if / f .(jc)> 0 for all i.Let 7=1 Our main result is the following formula for the restriction of ω to l (1.3) ω = yΓϊddπ* ΓΓ^ίLog/,.)+ lA .This we will derive as a corollary of another result which I will now describe: By [5] there is an intrinsic involution γ: X -• X which reverses