Motivic invariants and birational geometry of simple normal crossing degenerations

该项目的目的是开发一个新的框架的双有理类型和简单的正常计划的不变量，并应用这个框架来重新审视长期存在的基本问题，代数几何。这是通过三个步骤实现的。第一个关键因素是在简单的正态交叉方案之间引入双有理收缩的范畴，以将它们视为光滑的。在这个范畴中，取有理映射的极限，一个非常困难的经典问题，成为一个基本上正式的步骤，而注意力转移到新构造的范畴的性质。其次，我们调查新的不变量的简单正常的交叉计划，作为函子在这个双理性范畴。这一部分的目标包括解决最近和非常成功的不变量的分类问题，如motivic体积和对角分解，并提供一个新的motivic（通用）建设的限制混合霍奇结构。最后，我们给出了新框架在解决代数几何中的一些老问题，如Luroth问题上的应用。我们的目标是在合理性问题领域取得实质性进展，其中许多问题很容易制定，但至少在过去50年中没有得到解决。这是通过结合现有的退化方法，从光滑品种简单的正常交叉计划，与强大的新构造的不变量。

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.