Topological research of the theory of toric varieties

我们从拓扑学的角度发展了环面变分理论。环面变分理论认为，“环面变分”（代数几何中的一个对象）和“扇形”（组合学中的一个对象）之间存在一一对应关系。在我们的项目中，我们研究了“环面流形”或“环面轨道”，它们是环面变体的拓扑对应物，是比环面变体更宽的对象，并构建了这些扩展对象与扩展组合对象的对应关系，称为“多扇”。我们通信中的一个基本问题是几何上得到的多扇的特征化，我们完整地描述了环面轨道得到的多扇。此外，我们还从多扇的角度描述了环面流形的特征和t_y属。矩映射给出了另一个基本的对应关系。引入了多多边形的概念，并将凸多边形的Ehrhart多项式和khovanski - pukhlikov公式推广到多多边形。

We developed the theory of toric varieties from topological viewpoint. The theory of toric varieties says that there is a one-to-one correspondence between "toric varieties" (an object in algebraic geometry) and "fans" (an object in combinatorics). In our project, we studied "torus manifolds" or "torus orbifolds" which are topological counterparts to toric varieties and a wider object than that of toric varieties, and constructed a correspondence from those extended objects to an extended combinatorial object called "multi-fans". One of the fundamental problems in our correspondence is to characterize geometrically obtained multi-fans, and we completely characterized the multi-fans obtained form torus orbifolds. Moreover, we described signatures and T_y-genera of torus manifolds in terms of multi-fans. There is another fundamental correspondence given by moment maps. We introduced a notion of multi-polytopes, which appear as images of moment maps, and generalized Ehrhart polynomials and Khovanskii-Pukhlikov formula for convex polytopes to multi-polytopes.