Holomorphic Symplectic Varieties, Mirror Symmetry, and Cluster Algebras

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Hyper-Kähler manifolds, a type of geometric space with rich internal structure, are key geometric objects in algebraic geometry and differential geometry. These spaces play important roles in both mathematics and theoretical physics. The main goal of this research project is to construct new examples, or to prove that they do not exist. The research will investigate how hyper-Kähler manifolds degenerate, or break into simpler pieces. The investigator will first study how the known examples degenerate, to infer general patterns, and will then work to reverse the process to construct new examples. The work will make use of mirror symmetry, a remarkable correspondence between pairs of geometric spaces discovered by physicists. For hyper-Kähler manifolds, mirror symmetry can be understood quite explicitly. This provides information about degenerations of hyper-Kähler manifolds, for example, the combinatorial way in which the different pieces fit together.This award supports research at the interface between algebraic geometry and differential geometry on hyper-Kähler manifolds, which are important in geometry and theoretical physics. Only a few types of examples are known, and previous constructions have been based on classical algebraic geometry. The investigator will use new techniques coming from mirror symmetry, cluster algebras, and deformation theory to produce new examples, or alternatively, to discover tangible obstructions to their existence. In prior collaborative work, he developed a geometric interpretation of cluster algebras in terms of toric and birational geometry. These geometric spaces, called cluster varieties, are expected to form the building blocks of degenerations of hyper-Kähler manifolds. In subsequent collaborative work, the investigator studied mirror symmetry for cluster varieties. This involves piecewise linear geometric structures that can be used to understand how to glue cluster varieties together to form a hyper-Kähler manifold.