Categorical Kahler Geometry and Applications

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Birational geometry is a classical mathematical discipline whose roots go back to ancient Greece. Nevertheless, it still offers many difficult unsolved questions. The core part of this project is to tackle these questions with cutting-edge modern methods coming from the homological mirror symmetry program. Homological mirror symmetry is a deep geometric duality that originates in quantum field theory and has been used in studying novel phenomena and proving unexpected results in symplectic geometry suggested by algebraic geometry. This project aims to use homological mirror symmetry to introduce new applications of symplectic topology to algebraic geometry and to answer classical open questions in birational geometry. A postdoctoral fellow will be involved in various aspects of this research project.This project will use an approach based on categorical Kähler geometry. The most notable application of this approach is toward proving the non-rationality of generic, four-dimensional cubics, which is arguably the central problem in algebraic geometry. More specifically, a detailed study of the singularities of quantum D-modules produces a completely new type of birational invariant. This new invariant is a canonical decomposition of the cohomology of a four-dimensional cubic based on simultaneous use of both (algebraic and symplectic) sides of homological mirror symmetry. The example of four-dimensional cubics is only the tip of the iceberg. There are many other potential applications of this approach, for example to uniformization problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.