Geometry and Topology of Holomorphic Symplectic Varieties

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Algebraic geometry is a subject studying algebraic varieties, which are spaces described by solutions of polynomial equations. This subject started with the work of the ancient Greeks about ellipses, parabolas, and hyperbolas, and now it has grown to be a modern subject sharing many rich connections with physics and computer science. The research supported by this NSF award will focus on an important class of algebraic varieties, called the holomorphic symplectic varieties, that lie at the crossroads of geometry, representation theory, and mathematical physics. A better understanding of these varieties will bring deep insights in both algebraic geometry and its related fields.In this project, the PI will build connections between compact and noncompact holomorphic symplectic varieties with a focus on the P=W conjecture, which is a central conjecture connecting Hitchin's integrable systems and Hodge theory. Compact holomorphic symplectic varieties, which are higher dimensional generalizations of K3 surfaces, have been studied intensively for decades. A systematic way to compare compact and noncompact geometries, and to introduce new techniques to Hitchin systems from compact holomorphic symplectic varieties will bring fresh perspectives and fruitful outcomes to the P=W conjecture and related questions.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.