埃瓦尔德猜想与代数和辛格几何中的整数点
Ewald's Conjecture and integer points in algebraic and symplectic toric geometry
AI总结：
本文解决了几何中关于多面体整数点的开放问题，首次证明了埃瓦尔德猜想的广义情况，并探讨了辛格几何中轨道位移问题，同时证明了尼尔猜想在二维和高维情况下的结论。
摘要：
我们解决了几个关于辛几何和代数几何中多面体整数点的开放问题。在此方向上，我们给出了埃瓦尔德猜想（1988）关于对称整数点的单调格点多面体在任意维度中的广泛证明。我们还对埃瓦尔德猜想中出现的点集进行了渐近定量研究。然后我们将这项工作与辛格几何中轨道位移问题联系起来。最后，我们证明了在二维情况以及若干高维情况下的尼尔猜想（2009），这是埃瓦尔德猜想在光滑格点多面体上的推广。在此过程中，本文引入了两种新的多面体类别，这些多面体在研究埃瓦尔德猜想和辛位移性时自然出现：与奥达猜想相关的整洁多面体和深度单调多面体。
英文摘要：
We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.
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