Derived Categories, Hodge Theory, and Birational Geometry

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Algebraic geometry is the study of the geometric objects -- called algebraic varieties -- defined by systems of polynomial equations. A fundamental problem is to classify algebraic varieties, i.e. to determine when one can be transformed into another using algebraic functions. The main theme of this project is to study the classification problem using certain algebraic invariants (derived categories and Hodge structures), which can be thought of as sophisticated "linear approximations" to algebraic varieties. These invariants have connections to many fields, ranging from number theory to symplectic geometry and high energy physics. The project has three related parts. The first is to use Bridgeland stability conditions to prove results about the geometry and period mappings of Fano varieties; this relies on a newly developed notion of stability conditions in families, and the existence of noncommutative K3 surfaces in the derived categories of certain Fano varieties. The second part is to construct more examples of noncommutative K3 surfaces, and to further develop the theory of homological projective geometry (which gives a powerful tool for studying noncommutative varieties in general). The third part is to study geometric problems suggested by the first two parts, concerning the rationality of algebraic varieties and the construction of holomorphic symplectic varieties.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.