Motivic invariants and birational geometry of simple normal crossing degenerations

简单正态交叉退化的动机不变量和双有理几何

• 批准号: EP/Z000955/1
• 负责人: Evgeny Shinder
• 金额: $ 221.76万
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FZ000955%2F1
• 关键词: Motivic; invariants; birational; geometry; simple

The project is designed to develop a new framework of birational types and invariants of simple normal schemes, and to apply this framework to revisit long-standing fundamental problems in algebraic geometry. This is achieved in three steps. The first key ingredient is introducing the category of birational contractions between simple normal crossing schemes to treat them as if they were smooth. In this category taking limits of rational maps, a very difficult classical problem, becomes an essentially formal step, while the attention is shifted to the properties of the newly constructed category. Second, we investigate new invariants of simple normal crossing schemes, as functors on this birational category. The goals in this part include solving the problem of categorifying recent and very successful invariants such as the motivic volume and the decomposition of the diagonal, and providing a new motivic (universal) construction for the limiting mixed Hodge structure. Finally, we work out applications of the new framework to the old and difficult conjectures in algebraic geometry, such as the Luroth problem. We aim for a substantial progress in the area of rationality problems, where many questions are easily formulated, but have not been solved for at least the last 50 years. This is done by combining the existing degeneration methods, from smooth varieties to simple normal crossing schemes, with the powerful newly constructed invariants.

该项目旨在开发一个新的双有理类型和简单正规格式不变量的框架，并应用该框架重新审视代数几何中长期存在的基本问题。这是通过三个步骤实现的。第一个关键要素是在简单的正常交叉方案之间引入双有理收缩类别，以将它们视为平滑的。在这个范畴中，对理性映射的限制，一个非常困难的经典问题，基本上成为一个正式的步骤，同时注意力转移到新构建的范畴的属性上。其次，我们研究简单正态交叉方案的新不变量，作为这个双有理范畴的函子。这部分的目标包括解决最近非常成功的不变量（例如动机体积和对角线分解）的分类问题，并为极限混合 Hodge 结构提供新的动机（通用）构造。最后，我们研究了新框架在代数几何中古老而困难的猜想（例如卢罗斯问题）中的应用。我们的目标是在理性问题领域取得实质性进展，该领域的许多问题很容易提出，但至少在过去 50 年里尚未得到解决。这是通过将现有的退化方法（从平滑变异到简单的正态杂交方案）与强大的新构建的不变量相结合来完成的。