Topological research of the theory of toric varieties

我们从拓扑学的观点发展了复曲面簇的理论。复曲面簇理论认为，在“复曲面簇”（代数几何中的一个对象）和“扇形”（组合数学中的一个对象）之间存在一一对应的关系。在我们的项目中，我们研究了“环面流形”或“环面orbifolds”，这是拓扑对应的环面品种和一个更广泛的对象比环面品种，并构建了一个对应的扩展对象的组合对象称为“多扇”。我们的通信中的一个基本问题是几何上获得的多扇的特征，我们完全特征化的多扇从环面orbifolds。此外，我们还用多重扇描述了环面流形的签名和T_y-亏格。还有另一个基本的对应关系由矩映射给出。我们引入了多多面体的概念，它表现为矩映射的图像，并将凸多面体的Ehrhart多项式和Khovanskiii-Pukhlikov公式推广为多多面体。

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.