Hyperkähler Manifolds, Moduli Spaces, and Fano Varieties

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This project is focused on questions in algebraic geometry. This field of mathematics focuses on geometric spaces, called algebraic varieties, which can locally be described as the set of solutions of a system of polynomial equations in several variables. As such, the field lends itself to applications in computing and information as well as computer science and physics. One of the main aims of algebraic geometry is to classify algebraic varieties. One discrete invariant that can be used to distinguish distinct classes is curvature: varieties with positive curvature are called Fano, while varieties with zero curvature are called Calabi-Yau, and those with negative curvature are called general type. This project is focused on a particular class of Calabi-Yau varieties, which are called hyperkähler manifolds. The project uses moduli theory to explore relationships between hyperkähler manifolds and varieties that are Fano, varieties that are Calabi-Yau, and varieties that are general type. The project also supports the PI’s continued efforts and activities towards broadening participation among underrepresented groups.The project centers around three broad goals. (1) The PI aims to strengthen connections between hyperkähler manifolds and Fano varieties by formalizing geometric constructions associating a Fano variety to a hyperkähler manifold of K3 type. (2) Another goal is to advance the study of Lagrangian fibrations of hyperkähler manifolds by investigating which abelian varieties may arise as smooth fibers of a Lagrangian-fibered hyperkähler manifold. (3) The PI will expand the theory of moduli of hyperkähler manifolds by studying the geometry of such moduli spaces, in particular describing when moduli spaces of hyperkähler manifolds are of general type. In addition, the PI will co-organize various events with a view towards educating and training the next generation and growing broad participation in the mathematical sciences.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.