Kahler geometry of toric manifolds in symplectic coordinates

By M. Abreu. 2000

207 citations recorded by the source corpus.

A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $\phi:M\to\R^n$, a convex polytope $P=\phi(M)\subset\R^n$. In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-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 K\"ahler toric geometry, namely: explicit construction of extremal K\"ahler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/94edff23dd29c2da7b5e638b40e48280ea7d86f8

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Polar and coisotropic actions on Kähler manifolds (2000, background): “ince QP is not empty. Hence QP contains regular points; if q ∈ QP is regular, then we have the orthogonal splitting TqMP = Tq(Gq)⊕TqQP (with respect to gP ) and JP (TqQP ) = Tq(Gq). 11 Abreu shows in [Ab] that the complex manifolds (M,J) and (MP,JP ) are Gequivariantly biholomorphic. If φ is such a ”
Quantizing N = 2 multicenter solutions (2008, methodology): “It follows from the construction [45, 46] that K is the Legendre transform of g, i.”
Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, background): “Let ∆ be a Delzant polytope in Rn obtained as the convex hull of points in Zn (see [1]) It induces a triple (X ∆, L ∆,φ∆), where X ∆, is an n-dimensional complex compact projective manifold on which the complex torus C∗n acts effectively with an open dense orbit and (L ∆,φ∆) is an Hermitian positive”
Sasaki-Einstein Manifolds (2010, background): “2 extends earlier work of Guillemin [57] and Abreu [1] from the Kähler case to the Sasakian case.”
Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds (2006, methodology): “1 in [1]) we may assume that the complex structure obtained by the Delzant construction is the same with the complex structure of the Kähler cone C(S) of the Sasaki manifold S under consideration.”
Examples of asymptotically conical Ricci-flat Kähler manifolds (2008, background): “The Kähler metric can be seen as in [2] to be of the form”
Ricci-flat Kähler metrics on crepant resolutions of Kähler cones (2008, background): “The Kähler metric can be seen as in [1] to be of the form”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “De plus, en utilisant la dualité, nous confirmons aussi une conjecture de Donaldson sur les solutions du problème de valeurs au bord d’Abreu sur le corps convexe P dans le cas d’une mesure canonique donnée sur la frontière de P .”
Hamiltonian 2-forms in Kähler geometry, II Global Classification (2004, background): “According to Abreu [1, 2], when g0 is the canonical metric on (M,ω), these conditions are not only sufficient but necessary for the compactification of g.”
Black holes as effective geometries (2008, background): “In fact the manifold is a toric Kähler manifold and can thus be quantized using the technology of [88] [89].”