Algebraic and Arithmetic Geometry Seminar

Mirror symmetry for generalized Kummer varieties

by Justin Sawon (University of North Carolina at Chapel Hill, USA)

Aula Riunioni (Department of Mathematics)

Aula Riunioni

Department of Mathematics


The generalized Kummer variety $K_n$ of an abelian surface $A$ is the fibre of the natural map $\mathsf{Hilb}^{n+1}A\to \mathsf{Sym}^{n+1}A\to A$. Debarre described a Lagrangian fibration on $K_n$ whose fibres are the kernels of $\mathsf{Jac}C\to A$, where $C$ are curves in a fixed linear system in $A$.
In this talk we consider the dual of the Debarre system, constructed in a similar way to the duality between $\mathsf{SL}$- and $\mathsf{PGL}$-Hitchin systems described by Hausel and Thaddeus. We conjecture that these dual fibrations are mirror symmetric, in the sense that their (stringy) Hodge numbers are equal, and we verify this in a few cases. In fact, there is another isotrivial Lagrangian fibration on $K_n$. We can describe its dual fibration and verify the mirror symmetry relation in many more cases.