Abelian varieties associated to Clifford algebras

The Kuga-Satake construction is a construction in algebraic geometry which associates an abelian variety to a polarized K3-surface X. This abelian variety, A, is created from the Clifford algebra arising from the quadratic space H2(X, Z)/torsion with its natural cohomology pairing. Furthermore, there is an inclusion of Hodge structures H2( X, Q)hookrightarrow H1( A, Q) ⊗ H1( A, Q) relating the cohomology of the original K3-surface with that of the abelian variety. We investigate when this construction can be generalized to both arbitrary quadratic forms as well as higher degree forms. Specifically, we associate an abelian variety to the Clifford algebra of an arbitrary quadratic form in a way which generalizes the Kuga-Satake construction. When the quadratic form arises as the intersection pairing on the middle-dimensional cohomology of an algebraic variety Y, we investigate when the cohomology of the abelian variety can be related to that of Y. Additionally, we explore when families of algebraic varieties give rise to families of abelian varieties via this construction. We use these techniques to build an analogous method for constructing an abelian variety from the generalized Clifford algebra of a higher degree form. We find certain families of complex projective 3-folds and 4-folds for which an abelian variety can be constructed from the respective cubic and quartic forms on H2. The relations between the cohomology of the abelian variety and the original variety are also discussed.