Motivic invariants and birational geometry of simple normal crossing degenerations

该项目旨在开发一个简单正规格式的二元类型和不变量的新框架，并应用该框架来重新审视代数几何中长期存在的基本问题。这可以通过三步来实现。第一个关键因素是在简单的正常交叉方案之间引入两种收缩，将它们视为平滑的。在这个范畴中，求取有理映射的极限，一个非常困难的经典问题，成为一个本质上形式化的步骤，而注意力则转移到新构造范畴的性质上。其次，我们研究了简单正规交叉格式的新不变量，作为双范畴上的函子。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，如动机体积和对角线分解，并为限制混合Hodge结构提供一个新的动机（通用）结构。最后，我们研究了新框架在代数几何中旧的和困难的猜想中的应用，如Luroth问题。我们的目标是在理性问题领域取得实质性进展，在这个领域，许多问题很容易提出，但至少在过去的50年里没有得到解决。这是通过将现有的退化方法（从光滑变异到简单的正态杂交方案）与新构造的强大不变量相结合来实现的。

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.