Kähler geometry of toric manifolds in symplectic coordinates

Abstract

A theorem of Delzant states that any symplectic manifold (M, ω) of dimension 2n, equipped with an effective Hamiltonian action of the standard n-torus T n = R n /2πZ n , is a smooth projective toric variety completely determined (as a Hamiltonian T n -space) by the image of the moment map φ : M → R n , a convex polytope P = φ(M ) ⊂ R n . In this paper we show, using symplectic (action-angle) coordinates on P ×T n , how all ω-compatible toric complex structures on M can be effectively parametrized by smooth functions on P . We also discuss some topics suited for application of this symplectic coordinates approach to Kähler toric geometry, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

“…Starting from a Delzant polytope, we construct a symplectic toric manifold in Subsection 5.1 and a complex toric manifold in Subsection 5.2. We identify them according to a choice of symplectic potentials due to [Ab1,Ab2,Gu1,Gu2] in Subsection 5.3. We also review certain deformation of toric Kähler structures by changing symplectic potentials, which was introduced in [BFMN].…”

“…In this section we identify them, using symplectic potentials due to [Gu1,Gu2,Ab1,Ab2]. We also recall a certain deformation of toric Kähler structures due to [BFMN].…”

“…Then the principle can be understood naturally if there is a family {P Js } s∈[0,∞) of Kähler polarizations on M with P J 0 = P J which converges to P µ in the sense that there exists a basis {σ m s } m∈BS(Pµ) of H(P Js ) for each s ∈ [0, ∞) such that, for each m ∈ BS(P µ ), σ m s converges to a delta-function section supported on the Bohr-Sommerfeld fiber µ −1 (m) as s goes to ∞. In [BFMN], the authors carried out such a construction in the case of a non-singular projective toric variety by changing symplectic potentials, an important notion in the deformation theory of toric Kähler structures due to Guillemin [Gu1,Gu2] and Abreu [Ab1,Ab2].…”

Toric Kähler Metrics: Cohomogeneity One Examples of Constant Scalar Curvature in Action-Angle Coordinates

“…It is an immediate extension to our possibly non-compact setting of the compact version proved in [2]. To properly state it we need the following definition.…”

“…In particular, when n = 2, on all Hirzebruch surfaces. The main goal of these notes is to present Calabi's general family in local symplectic actionangle coordinates, using the set-up of [1,2] for toric Kähler geometry, and show that it actually contains other interesting cohomogeneity one Kähler metrics as particular cases (see also [22]). These include:…”