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植物类型和简单普通方案的不变式的框架，并应用此框架来重新审视代数几何形状的长期基本问题。这是三个步骤实现的。第一个关键要素是在简单的正常越过方案之间引入异性收缩类别，以将其视为光滑。在采用理性地图的限制的类别中，一个非常困难的经典问题是本质上是正式的一步，而注意力转移到了新建类别的属性。其次，我们研究了这个birational类别的功能因素，研究了简单正常越过方案的新不变性。本部分的目标包括解决近期和非常成功的不变性的问题，例如动机量和对角线的分解，并为限制混合霍奇结构提供新的动机（通用）结构。最后，我们将新框架的应用程序应用于代数几何形状（例如Luroth问题）中的旧构成和困难的猜想。我们的目标是在理性问题领域取得重大进展，在这些方面很容易提出许多问题，但至少在过去的50年中没有得到解决。这是通过将现有的变性方法与功能强大的新建不变式结合到现有的变性方法来完成的。