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Kähler geometry of toric manifolds in symplectic coordinates. (English) Zbl 1044.53051
Eliashberg, Yakov (ed.) et al., Symplectic and contact topology: Interactions and perspectives. Papers of the workshop on symplectic and contact topology, quantum cohomology, and symplectic field theory, Montreal and Toronto, Canada, March–April 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3162-3/hbk). Fields Inst. Commun. 35, 1-24 (2003).
A triple \((M,J,\omega)\), where \(M\) is a manifold of real dimension \(2n\), \(J\) an almost complex structure, and \(\omega\) a 2-form, is a Kähler manifold if \((M,J)\) is a complex manifold (i.e. \(J\) is integrable), \((M,\omega)\) is a symplectic manifold (i.e. \(\omega\) is closed and nondegenerate), and the form \(\rho(x,y)=\omega(x,Jy)\) is symmetric and positive definite. If the complex structure is fixed then any symplectic structure compatible with it has locally the form \(\omega=2i\partial\bar\partial f\), where \(f\) is some real function. On the other hand, no effective parametrization of complex structures compatible with a fixed symplectic form is known.
In this paper, the problem is considered for toric manifolds. A (complex) toric manifold is a compact complex manifold \(M\) endowed with an effective holomorphic action of a complex torus that has an open dense orbit. A symplectic \(2n\)-manifold \(M\) is toric if it is endowed with a Hamiltonian action of the real \(n\)-torus which is free on some dense and open subset. The image of the moment mapping on \(M\) is a convex polytope \(P\) in \({\mathbb R}^n\) (these polytopes were described by Delzant). A Kähler toric manifold \((M,J,\omega,\tau)\), where \(\dim_{\,\mathbb C} M=n\), is a compact Kähler manifold \((M,J,\omega)\) equipped with an effective Hamiltonian holomorphic action \(\tau\) of the real \(n\)-torus. The polytope \(P\) uniquely determines the toric symplectic structure and the toric complex structure but the toric Kähler structure compatible with them is not unique (for example, this is true for circle actions on \(S^2\)).
More precisely, let \(P^o\) be the (nonempty) interior of \(P\), and \(\omega=\sum_{k=1}^ndx_k\wedge dy_k\) be the symplectic form on \(P^o\times{\mathbb T}^n\), and where \((x,y)\) are natural coordinates for the product \(P^o\times{\mathbb T}^{n}\), and where \({\mathbb T}^{n}\) is realized as \({\mathbb R}^n/{\mathbb Z}^n\). Then the structure of a Kähler toric manifold can be defined by \(J(x,y)=(-G^{-1}y,Gx)\), where \(G\) is the Hesse matrix of some function \(g\) that does not depend on \(y\). If the standard complex structure \((u,v)\mapsto(-v,u)\) is fixed then the Kähler toric structure can be defined by the symplectic form \(\omega=2i\partial\bar\partial f\) for some \(f\) that does not depend on \(v\). Two functions \(f\) and \(g\) define equivalent structures if they correspond by the Legendre transform: \(f(u)+g(x)=\sum_{k=1}^n {{\partial f}\over{\partial u_k}}{{\partial g}\over{\partial x_k}}\), where \(x={{\partial f}\over{\partial u}}\) (or \(u={{\partial g}\over{\partial x}}\)). The condition that the manifold is smooth determines the type of singularities near the boundary of \(P\): \(g=g_P+h\), where \(h\) is smooth on \(P\) and \(g_P(x)={1\over2}\sum l_k(x)\log l_k(x)\), where the sum is taken over the \((n-1)\)-faces of \(P\), \(l_k(x)=\langle x,\mu_k\rangle-\lambda_k\) are affine functions defining these faces by \(l_k(x)=0\) (\(\mu_k\) are orthogonal to them, \(| \mu_k| =1\), and \(l_k(x)>0\) on \(P^o\)).
Besides, the paper contains a discussion of extremal Kähler metrics, spectral properties of toric manifolds, and connections between Kähler geometry and combinatorics of polytopes.
For the entire collection see [Zbl 1009.00020].

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
53D20 Momentum maps; symplectic reduction