Ljudmila Kamenova (Stony Brook University),
Giovanni Mongardi (University of Bologna),
Alexei Oblomkov (UMass Amherst)
Hyperkahler manifolds are higher-dimensional generalizations of K3 surfaces, and as such they are a perfect testing ground for central conjectures in algebraic geometry, like the Hodge conjecture, the Tate conjecture, Grothendieck’s standard conjectures, Bloch-Beilinson conjectures, etc. Exciting new insight has been obtained in recent years, but many new and old questions remain open. Recent developments put hyperkahler manifolds at a central place in the interaction between complex geometry, arithmetic algebraic geometry, derived category theory, and more. Bringing together people working on various specific problems of this sort and experts interested in the general geometry of hyperkahler varieties should trigger further results.
We intend to focus on a detailed understanding of the general conjectures for special classes of hyperkahler manifolds, e.g., those of K3[n]-type, which could lead to new insight into the conjectures in general. Of very high recent interest are also irreducible symplectic varieties, which are a singular analogue of hyperkahler manifolds and share with them most of the interesting properties. Studying these cases is relevant in order to understand general properties of hyperkahler geometry. As an example, general results linking their Hodge structure with their geometry have been recently proven by Bakker and Lehn, and in the case of orbifolds a twistor family structure was constructed by Menet.
This program also has an assocaited workshop: Hyperkahler quotients, singularities, and quivers: January 30-February 3, 2023.