Integrable bodies in odd-dimensional spaces

代数可积的光滑凸体

Algebraically Integrable Smooth Convex Bodies

该研究证明奇数维n≥3的C^∞边界凸体代数可积当且仅为椭球，解决光滑凸体的阿诺德问题，结合瓦西里耶夫定理给出n≥2的分类，还推广了奇数维半代数边界凸体的相关结论。

Integrable systems, toric degenerations and Okounkov bodies

Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions H_1, ... H_n on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Moreover, we show that in many cases, we c…

Lacunas and local algebraicity of volume functions

The volume cut off by a hyperplane from a bounded body with smooth boundary in $R^{2k}$ never is an algebraic function on the space of hyperplanes: for k=1 it is the famous lemma XXVIII from Newton's Principia. Following an analogy of these volume functions with the solutions of hyperbolic PDE's, we study the local version of the same problem: can such a volume function coincide with an algebraic one at least in some domains of the space of hyperplanes, intersecting the body? We prove some homological and geometric obstructions to this integrability property. Based on these restrictions, we fi…

C型Richardson边界与奇维射影空间的Newton-Okounkov退化

Type C Richardson Boundaries and Newton-Okounkov Degenerations of Odd-Dimensional Projective Space

该研究对比奇维射影空间的两种镜像理论退化图景，计算C型边界与对应Laurent多项式，构造Newton–Okounkov体并验证半群等同性，还给出连接两类边界的秩一权退化。