Boundary components of Mumford–Tate domains

Abstract

We study certain spaces of nilpotent orbits in the Hodge domains introduced by [GGK1], and treat a number of examples. More precisely, we compute the Mumford-Tate group of the limit mixed Hodge structure of a generic such orbit. The result is used to present these spaces as iteratively fibered algebraic-group orbits in a minimal way. 1 arXiv:1210.5301v2 [math.AG] 8 Apr 2015 9 These are stated for the rank-one case, cf. §2 (where B(N ) and the {I p,q } are defined); they have obvious generalizations replacing N with σ. 10 In [GGK1] (cf. Example VI.B.15), it was discovered that (the finitely many) connected components of the subset of a period domain comprising Hodge structures with Mumford-Tate group contained in a given (Mumford-Tate) subgroup of Aut(V, Q), need not have the same generic Mumford-Tate group.

Kato-Usui partial compactifications over the toroidal compactifications of Siegel spaces

“…A similar result is given by Kerr and Pearlstein [KP,Proposition 7.4], where they show that certain boundary components are holomorphically fibered over Shimura varieties. However the situation of the above theorem is different from their setting.…”

“…In addition, Kerr and Pearlstein showed a rigidity theorem for variations of Hodge structure. By [KP,Proposition 10.6], a limiting mixed Hodge structure determines a variation of Hodge structure. Therefore limit points in boundary components have importance for variations of Hodge structure.…”

Variations of Hodge structure and orbits in flag varieties

“…As a consequence, j g p,q = j g p ∩ j g q . 19 In the notation of [34], the domain Dj is denoted D(Nj ), and the Hodge structure ϕj,• by ϕ split .…”

“…One thereby arrives at maximal expected dimensions for images of algebro-geometric period maps in any Γ\D. In the asymptotic direction, a good understanding of abelian nilpotent cones in g R allows for a classification of boundary components parametrizing all possible limiting mixed Hodge structures (LMHS) [32,34]. This leads to precise predictions for the degeneration of varieties (and GIT-type boundary strata of relevant moduli spaces), as well as a means of attacking Torelli-type problems [14,52].…”

Principal Hodge representations

“…, 1). Examples of such Hodge structures include the cohomology group H 1 (X, Q) of an elliptic curve, the cohomology group H 3 (X, Q) of a mirror quintic variety [3], and the weight component W 6 H 6 (X, Q) of the cohomology group of a general fibre of a family of quasi-projective 6-folds studied by Dettweiler and Reiter [2,6]. More precisely, the article addresses the question: what are the Hodge representations ( §2.1) with Hodge numbers h = (1, 1, .…”