Kuga varieties, K3 surfaces and the Kuga-Satake construction

Kuga varieties are a natural generalisation of universal families of abelian varieties. This thesis
describes the candidate’s work on the geometry of some types of Kuga varieties. In Part I, by
considering a special kind of Kuga varieties resulting from the Kuga-Satake construction, we construct
an explicit map from a moduli space of K3 surfaces of Picard rank 14 to a moduli space of polarised
abelian 8-folds with totally definite quaternion multiplication. This is a geometric interpretation of
an exceptional coincidence between locally symmetric spaces of type II_4 and type IV_6. In Part II,
we study the n-fold Kuga varieties associated to the moduli space of (1, p)-polarised abelian surfaces
with canonical level structure for prime p at least 3, and compute their Kodaira dimensions for all
but 27 possible combinations of (n, p).