# Kahler geometry of toric manifolds in symplectic coordinates

@article{Abreu2000KahlerGO,
title={Kahler geometry of toric manifolds in symplectic coordinates},
author={Miguel Abreu},
journal={arXiv: Differential Geometry},
year={2000}
}
• M. Abreu
• Published 19 April 2000
• Mathematics
• arXiv: Differential Geometry
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…
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## References

SHOWING 1-10 OF 23 REFERENCES

• Mathematics
• 1999
A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or
A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler
This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a
1. Let (X, ω) be a compact connected 2w-dimensional manifold, and let (1.1) τ: T -+Όifί(X, ω) be an effective Hamiltonian action of the standard w-torus. Let φ: X —> R be its moment map. The image,
Given a compact, complex manifold M with a Kahler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊΩ of all Kahler metrics in M in that class. To
• Mathematics
• 1994
1. Statement of results Let M be a compact manifold endowed with a Riemannian metric. The spectrum of the Laplacian, A, acting on functions form a discrete set of the form {0 < ),~ < 22 < �9 �9 �9 <
• Mathematics
• 2002
A theorem of J. Hersch (1970) states that for any smooth metric on S2, with total area equal to 4π, the first non‐zero eigenvalue of the Laplace operator acting on functions is less than or equal to
On utilise la methode de continuite pour demontrer l'existence de metriques de Kahler-Einstein pour des varietes de Kahler de fibre linge positif anticanonique sous l'hypothese additionnelle de
These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to