Arithmetical compactification of mixed Shimura varieties. (English) Zbl 0748.14007

Pink, Richard

Bonner Mathematische Schriften 209. Bonn: Univ. Bonn, Math.-Naturwiss. Fak. xvii, 340 p. (1989 ).

The author studies mixed Shimura-varieties. Recall that usual Shimura- varieties are quotients of bounded symmetric domains under arithmetic groups. They are not just analytic spaces but have an algebraic structure, can be compactified in various ways and are defined over number-fields. The simplest example is the moduli-space of elliptic curves. Mixed Shimura-varieties are certain fibre-spaces over these, like the total space of the universal elliptic curve. They also appear naturally in the boundary of toroidal compactifications of usual (=pure) Shimura-varieties. They classify mixed Hodge-structures.
This thesis defines them, constructs toroidal compactifications, and proves that they are algebraic and can be defined over number-fields.

Reviewer: G.Faltings (Princeton)

MSC:

14G35	Modular and Shimura varieties
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14D07	Variation of Hodge structures (algebro-geometric aspects)

Keywords:

mixed Shimura-varieties; toroidal compactifications; mixed Hodge structure