Abstracts for Seminar Algebraic Geometry (SAG)

For any complex algebraic group G, there is a complete ind-variety X parametrising connected subgroups of G and a universal family. I present this construction, where the semi-continuity of certain discrete invariants of Lie algebras play an important role.

Given a smooth projective $3$-fold $Y$, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing $1$-cycles in $Y$ to the intermediate Jacobian $J(Y)$. We consider in this talk the existence of families of $1$-cycles in $Y$ for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When $Y$ itself is uniruled, we relate this property to the existence of an integral homological decomposition of the diagonal of $Y$.

I will report on joint work with Bert van Geemen concerning moduli of principally polarised abelian fourfolds with an automorphism of order five. This moduli problem gives rise to a compact Shimura curve and a Shimura surface. I will explain a new technique to exhibit explicit projective models for them. The Shimura varieties possess a rich arithmetic and geometric structure that will be described in detail.