Toric geometry of convex quadrilaterals

Toric Pluripotential Theory

We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through the integrability properties of its Legendre transform. We characterize Log-Lipschitz convex functions on the Delzant polytope, showing that they correspond to toric qpsh functions which satisfy a certain exponential integrability condition. In the particular case of dimension one…

A Note on Toric Contact Geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant's theorem due to Lerman and Tolman [LT] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

Pitot四边上的极小曲面

Minimal surfaces over the Pitot quadrilaterals

本文提出了一种构造Pitot四边上Scherk型极小图的方法，通过调和映射和Weierstrass数据，得到具有特定高斯映射和高度函数行为的极小图，并证明了其曲率最大化性质。

Conical Calabi-Yau metrics on toric affine varieties and convex cones

It is shown that any affine toric variety Y, which is Q-Gorenstein, admits a conical Ricci flat Kahler metric, which is smooth on the regular locus of Y. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of Y. The case when the vertex point of Y is an isolated singularity was previously shown by Futaki-Ono-Wang. The proof is based on an existence result for the inhomogeneous Monge-Ampere equation in real Euclidean space with exponential right hand side and prescribed target given by a proper convex convex, combined with transversal a priori estimat…

Generalized toric varieties for simple non-rational convex polytopes

We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a complex quasitorus. We associate to each simple polytope, rational or not, a family of complex quasifolds having same dimension as the polytope, each containing a dense open orbit for the action of a suitable complex quasitorus. We show that each of these spaces M is diffeomorphic to one of the symplectic quasifolds defined in http://arXiv.org/abs/math:SG/9904179,…

环面几何中的非有理多面体与扇

Nonrational polytopes and fans in toric geometry

本研究在环面几何框架下考察非有理凸多面体与非有理扇的概念，梳理该领域最新进展及相互关联，为相关研究提供参考。

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different toric Kähler polarizations by taking the time-$\sqrt{-1} s$ Hamiltonian "flow" of strongly convex functions on the moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of $X_{P}$ in the (singular) real toric polarization. Recall that $X_{P}$ has an open dens…

Minimizing weak solutions for calabi's extremal metrics on toric manifolds

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

The volume of Kähler-Einstein Fano varieties and convex bodies

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser-Trudinger type inequalities for Stein domains and also leads to criticality results for mean…

Toric Varieties vs. Horofunction Compactifications of Polyhedral Norms

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geometry, normed spaces in convex analysis, and horofunction compactifications in metric geometry are directly and explicitly related.

Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that (X,D) admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Zhou-Wang concerning the case when…

Toric Hyperkahler Varieties

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanle…