Motivic invariants and birational geometry of simple normal crossing degenerations

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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.