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Good structures in higher dimensional birational geometry

高维双有理几何中的良好结构

基本信息

批准号：
239673722
负责人：
Professor Dr. Vladimir Lazic
金额：
--
依托单位：
Mathematisches Institut
依托单位国家：
德国
项目类别：
Independent Junior Research Groups
财政年份：
2013
资助国家：
德国
起止时间：
2012-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/239673722?language=en
关键词：
Good structures higher dimensional birational

项目摘要

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.

这项提议的目标是在高维变化的几何学方面取得突破性进展。良好的最小模型：最小模型程序的目标是对高维代数簇进行分类，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粗略的了解，该方案在1980年代仅在维度3中完全解决。最近，该领域取得了惊人的进展，这在一定程度上要归功于我和我的合著者。然而，该计划仍远未完成，留下的主要悬而未决的问题是大量的猜测和良好模型的存在。这个项目的目的是通过应用我和我的合著者开创的最新技术来证明这两个猜想，并建立一个关于多重正则型的新的推广结果。Calabi-Yau流形：Calabi-Yau流形是流形中最重要的一类，它的研究是出了名的困难。关于Calabi-Yau的K？Ahler锥的结构和有理曲线的存在性的基础性工作是在20世纪90年代完成的。Morison和Kawamata的一个重要且极其困难的圆锥猜想，在镜像对称性的推动下，预言了Calabi-Yau的nef和可动圆锥，隐约地说，接近于有理多面体。这个猜想意味着在Picard数为2的Calabi-Yau三重曲线上存在有理曲线。我计划证明这个猜想适用于小Picard秩三重Calabi-Yau曲线，这将是迄今为止最大的突破。该方法使用了我和我的合著者介绍的最新技术，并将其归结为积极特征。