Good structures in higher dimensional birational geometry

本提案的目标是在高维品种的几何上取得突破性进展。良好的最小模型：最小模型程序的目的是对高维代数变量进行分类，推广曲线和曲面的分类。分类的目的是对射影流形的结构有一个大致的了解，该方案在20世纪80年代才在3维中完全解决。最近，这一领域取得了惊人的进展，部分归功于我和我的合著者。然而，该方案还远未完成，遗留的主要问题是大量的猜想和是否存在良好的模型。这个项目的目标是通过应用我和我的合作者开创的最新技术来证明这两个猜想，并通过建立一个新的多元形式的扩展结果。Calabi-Yau流形：Calabi-Yau流形是流形中最重要的一类，众所周知，它们很难研究。关于Calabi-Yau的K′ahler锥的结构和有理曲线的存在性的基础工作是在20世纪90年代完成的。Morrison和Kawamata在镜像对称的推动下提出了一个重要而又极其困难的锥体猜想，预言了Calabi-Yau的网状锥体和活动锥体，模糊地说，接近于理性多面体。该猜想暗示了Picard数为2的Calabi-Yau三折上有理曲线的存在性。我计划用Calabi-Yau三倍的小皮卡德秩来证明这个猜想，这将是迄今为止最大的突破。该方法使用了我和我的合作者最近介绍的技术，并将其还原为积极的特征。

The goal of this proposal is to make ground-breaking progress in geometry of higher dimensional varieties. Good minimal models: The aim of the Minimal Model Program is to classify higher dimensional algebraic varieties, generalising the classification of curves and surfaces. The purpose of the classification is to give a rough understanding of the structure of projective manifolds, and the programme was completely resolved only in dimension 3 in the 1980s. Recently there has been spectacular progress in the field, partially due to me and my coauthors. However, the programme remains far from being complete, and the main open problems left are Abundance conjecture and existence of good models. The goal of this project is to prove these two conjectures by applying recent techniques pioneered by me and my coauthors, and by establishing a new extension result for pluricanonical forms. Calabi-Yau manifolds: Calabi-Yau manifolds represent one of the most important classes of manifolds, and they are notoriously difficult to study. Foundational work on the structure of the K¨ ahler cone of a Calabi-Yau and the existence of rational curves was done in the 1990s. An important and extremely hard Cone conjecture of Morrison and Kawamata, motivated by mirror symmetry, predicts that the nef and movable cones of a Calabi-Yau are, vaguely speaking, close to being rational polyhedral. The conjecture implies existence of rational curves on Calabi-Yau threefolds with Picard number 2. I plan to prove this conjecture for Calabi-Yau threefolds of small Picard rank, which would be the biggest breakthrough to date. The approach uses recent techniques introduced by me and my coauthors, and reduction to positive characteristic.