Alexei Piskunov: Geometry of Relative Compactified Jacobians on Bielliptic Surfaces

Abstract:

Moduli spaces of sheaves on surfaces of Kodaira dimension zero often reflect deep geometric structures inherited from the underlying surface. While K3 and Abelian surfaces have been extensively studied in this context, bielliptic surfaces have remained less explored due to the added complexity from their torsion canonical bundles and infinite fundamental groups.

In this talk, I will focus on the moduli spaces of pure dimension one semistable sheaves (also called relative compactified Jacobians) on bielliptic surfaces. After describing their construction and basic properties, I will present results on their topology and geometry, including explicit computations of the first and second Betti numbers. I will also describe the structure of the Albanese map and show that its fibers are Calabi–Yau varieties up to finite étale cover. These results extend known phenomena from the K3, Abelian and Enriques cases to the bielliptic setting.