カラビ・ヤウ多様体の錐予想について

Calabi-Yau变数在包括代数几何在内的许多数学分支中扮演着重要的角色。Calabi-Yau变种的镜像对称性启发了D·莫里森提出锥形猜想。Y.Kawamata提出了一种精化方法，其动机源于二次几何。在这个方向上，除了两篇已发表的论文外，我还完成了一篇预印本。双态自同构群的研究不仅与锥猜想有关，而且具有独立的意义。一个吸引人的领域是从几何和算术两个角度来看的双态自同构的动力学。最近，S.Kawaguchi和J.H.Silverman提出了一个猜想，该猜想将占优有理自映射的动态度和算术度联系起来。这一猜想对于态射的某些情况是已知的，但对于任意有理映射则鲜为人知，因为高度函数缺乏函数性。在这个方向上，除了发表了两篇论文外，我还完成了两份预印本。

Calabi-Yau varieties play a prominent role in many branches of mathematics including algebraic geometry. The mirror symmetry of Calabi-Yau varieties enlightened D. Morrison to formulate the cone conjecture. Y. Kawamata proposed a refinement whose motivation originated from birational geometry. In this direction, besides two published papers, I completed one preprint.The study of the birational automorphism groups is not only interrelated with the cone conjecture but also of independent interest. One attractive area is the dynamics of birational automorphisms from both geometric and arithmetic viewpoints. Recently, S. Kawaguchi and J. H. Silverman proposed a conjecture which connects dynamical and arithmetic degrees of dominant rational self-maps. This conjecture is known for some cases of morphisms, but is much less known for arbitrary rational maps due to the lack of functoriality of height functions. In this direction, besides two published papers, I completed two preprints.