Topological research of the theory of toric varieties

我们从拓扑学的角度发展了环簇理论。环变元理论认为，“环变集”(代数几何中的对象)和“扇子”(组合学中的对象)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“环面流形”，它们是环面簇的拓扑对应，并且是比环面簇更广泛的对象，并构造了从这些扩展对象到称为“多扇”的扩展组合对象的对应。我们对应的一个基本问题是刻画几何上得到的多扇，我们完全刻画了由环面上的多扇得到的多扇。此外，我们还用多扇的形式刻画了环面流形的特征和Ty-亏格。矩图还给出了另一种基本对应关系。我们引入了多重多面体的概念，并将Ehrhart多项式和Khovanskii-Pukhlikov公式推广到多重多面体。

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.