A brief survey about moment polytopes of subvarieties of products of Grassmanians

2019 ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Moment Polytopes

scholarly journals Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 102
Author(s):  
Jae-Hyouk Lee ◽  
Kyeong-Dong Park ◽  
Sungmin Yoo
Keyword(s):  
Einstein Metrics ◽  
Homogeneous Spaces ◽  
Picard Number ◽  
Spherical Varieties ◽  
Symmetric Varieties ◽  
Moment Polytope ◽  
Kähler Einstein Metrics ◽  
Moment Polytopes

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure.

scholarly journals REMARKS ON INVARIANTS OF HAMILTONIAN LOOPS

2010 ◽  
Vol 02 (03) ◽  
pp. 277-325 ◽  
Author(s):  
EGOR SHELUKHIN
Keyword(s):  
Maslov Index ◽  
Universal Cover ◽  
Symplectic Manifolds ◽  
Linear Functions ◽  
Torus Action ◽  
Hamiltonian Diffeomorphisms ◽  
Toric Manifolds ◽  
Mixed Action ◽  
Nonlinear Maslov Index ◽  
Moment Polytopes

In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.

scholarly journals A Note on Toric Varieties Associated with Moduli Spaces

2011 ◽  
Vol 54 (3) ◽  
pp. 561-565
Author(s):  
James J. Uren
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Toric Variety ◽  
Toric Varieties ◽  
Pants Decomposition ◽  
Basic Facts ◽  
The Moment ◽  
Moment Polytopes

AbstractIn this note we give a brief review of the construction of a toric variety coming from a genus g ≥ 2 Riemann surface Σg equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety DMg , the so-called Delzant model of moduli space, for each genus g. We conclude this note with some basic facts about the moment polytopes of the varieties . In particular, we show that the varieties DMg constructed by Tyurin, and claimed to be smooth, are in fact singular for g ≥ 3.

scholarly journals Efficient Algorithms for Tensor Scaling, Quantum Marginals, and Moment Polytopes

2018 ◽  
Author(s):  
Peter Burgisser ◽  
Cole Franks ◽  
Ankit Garg ◽  
Rafael Oliveira ◽  
Michael Walter ◽  
...  
Keyword(s):  
Efficient Algorithms ◽  
Moment Polytopes

scholarly journals Toric degenerations of integrable systems on Grassmannians and polygon spaces

2014 ◽  
Vol 214 ◽  
pp. 125-168
Author(s):  
Yuichi Nohara ◽  
Kazushi Ueda
Keyword(s):  
Potential Function ◽  
Integrable System ◽  
Linear Transformation ◽  
Integrable Systems ◽  
Piecewise Linear ◽  
Lagrangian Torus ◽  
Toric Degenerations ◽  
Piecewise Linear Transformation ◽  
The Moment ◽  
Moment Polytopes

AbstractWe introduce a completely integrable system on the Grassmannian of 2-planes in ann-space associated with any triangulation of a polygon withnsides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

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