Algebra and Geometry on Algebraic Varieties

近几十年来，代数变异的研究非常活跃。代数变体本身及其结构的可育性一直吸引着代数和几何学者的关注。此外，从共形场论和Calabi-Yau流形的研究中可以看出，它与包括物理在内的各个领域的关系越来越密切。在本课题中，我们关注了代数变量上群的作用之间的相互关系、Kahler流形的Hodge理论和周期映射、代数变量上的各种模空间、算术变量上的共形场理论、k理论和数论，在代数变量的研究上取得了很大的进展。我们组织了几次会议，设计相关领域之间的紧密沟通，并派出项目成员参加相关会议。我们可以得到以下优秀的结果：抛物型稳定轮轴模空间的构造及其结构的研究与应用，可能是奇异的射影格式上稳定轮轴模空间的构造，从数学角度看的共形场理论，Calabi-Yau流形的构造、变形与镜像对称性，modelweil格的发展与应用，K3曲面的研究与应用，一般型曲面的存在性问题，Mori理论的发展。

In the last few decades the study on algebraic varieties has been very active. Not only algebraic varieties itself but also the fertility of the structures over algebraic varieties are attracting the scholars of algebra and geometry. Moreover, as we can see in the research of the conformal field theory and the Calabi-Yau manifolds, the relationship with various fields including physics is getting closer. In this project, paying attention to the interrelation among actions of groups on algebraic varieties, Hodge theory and period maps of Kahler manifolds, various moduli spaces on algebraic varieties, conformal field theory on arithmetic varieties, K-theory and number theory, we tried to make great progress in studying algebraic variety, In addition to individual studies in the neighborhoods of investigators, we organized several conferences to design close communication between related fields and sent members of the project to relevant conferences.We could get the following excellent results : construction of the moduli space of parabolic stable sheaves, study and applications of its structure, construction of the moduli space of stable sheaves on prejective schemes that may be singular, conformal field theory from the mathematical viewpoint, constructions, deformations and mirror symmetries of Calabi-Yau manifolds, development and applications of Model-Weil lattices, study and applications of K3 surfaces, existence prpblem of the surfaces of general type, development of Mori theory.