The space of compatible full conditionals is a unimodular toric variety

环面Deligne-Mumford栈的对角线消解

A Resolution of the Diagonal for Toric Deligne-Mumford Stacks

本文通过形变Bayer-Popescu-Sturmfels胞腔复形，给出光滑环面簇对应环面Deligne-Mumford栈的族形式对角线消解，还构造了有限阿贝尔群商对应环面Deligne-Mumford栈的对角线消解。

破碎环面簇与胞腔相容层

Broken Toric Varieties and Cell-Compatible Sheaves

本文利用常值层到多面体复形的导出前推研究破碎环面簇的上同调，证明Deligne型分解定理和谱序列退化，并给出部分情形的Betti数显式公式。

Tannakian classification of equivariant principal bundles on toric varieties

Let $X$ be a complete toric variety equipped with the action of a torus $T$ and $G$ a reductive algebraic group, defined over an algebraically closed field $K$. We introduce the notion of a compatible $Σ$--filtered algebra associated to $X$, generalizing the notion of a compatible $Σ$--filtered vector space due to Klyachko, where $Σ$ denotes the fan of $X$. We combine Klyachko's classification of $T$--equivariant vector bundles on $X$ with Nori's Tannakian approach to principal $G$--bundles, to give an equivalence of categories between $T$--equivariant principal $G$--bundles on $X$ and certain…

关于在辛奇点中刻画环形超kähler流形 II

Towards a characterization of toric hyperkähler varieties among symplectic singularities II

本文在前文基础上回答了问题5.10，证明了在特定条件下，锥形辛流形与环形超kähler流形存在equivariant代数同构。

Cluster algebras and Poisson geometry

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.

On toric varieties and algebraic semigroups

The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a proof of the fact that every separated toric variety may be constructed from a certain fan in a Euclidean space. To our best knowledge, this proof differs essentially from the ones which can be found in the literature.

Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,ω)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\bC P^r\times \bC P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\bC P^r \times \bC P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $ω$. In this paper we extend these results by considerin…

Standard monomial theory and toric degenerations of Richardson varieties in flag varieties

We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richard…

关于在辛奇点中刻画环面超凯勒簇

Towards a characterization of toric hyperkähler varieties among symplectic singularities

本文证明：具有射影辛解消且容许与锥形作用相容的$n$维环面有效哈密顿作用的锥形辛簇，必为环面超凯勒簇$Y(A,0)$（$A$幺模），且存在等变同构保持矩映射与中心。

复环面簇的链接束与相交空间

Link Bundles and Intersection Spaces of Complex Toric Varieties

本文计算了复三维环面簇相交空间的有理同调，发现其与相交同调不同，不具有组合不变性，从而保留了关于定义扇形的更精细信息。

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,…

超曲面的周期积分：热带几何方法

Period integrals of hypersurfaces via tropical geometry

本文利用热带几何方法计算环面簇中一参数复超曲面的周期积分渐近，并给出 d=1 时极限处的极化对数 Hodge 结构。

素数模一个（弯）同余的几何分类

Geometric classification of primes modulo a (bend) congruence

本文研究了素数同余的几何分类，通过四种应用扩展了现有结果，包括同余的强零点定理、取消性证明、热带函数半环建立及多面体闭包描述。

组合点击反应设计中的代数簇与理想理论

Algebraic Varieties and Ideal Theory in Combinatorial Click-Reaction Design

通过交换代数研究兼容性约束下的组合化学组装，构造组装理想并证明其零维且根式，刻画可逆反应三元组，应用于生物正交点击化学得到30个可行解和最大正交计划数4。