Compactifications of locally symmetric varieties

On the rigid cohomology of certain Shimura varieties

“…That this set is finite follows from theorem II.4.6 and the remark (ii) at the end of section II.4.1 of [3]. …”

Naive boundary strata and nilpotent orbits

“…It is known when D is a Siegel space H g (cf. [10,7]) that smooth toroidal compactifications (as in [3]) are obtained by adding in quotients of our B(σ)'s (parametrizing σ-nilpotent orbits), and as we shall see this holds more generally. The Baily-Borel compactification, on the other hand, is obtained by using Γ-invariant sections of K ⊗M D (for some M 0) to embed X in a projective space, and then taking (Zariski or analytic) closure.…”

“…We assume that the Hermitian form H(v, w) := −2iQ(v,w) on V + has signature (2, 1), so that the projectivization of those v ∈ V + with H(v, v) < 0 yields a (Picard) 2-ball B ⊂ P(V + ). This parametrizes Hodge structures on V with Hodge numbers (3,3), and a nontrivial involution (with eigenspaces V + , V + ). Alternatively, one can consider Carayol's nonclassical domain D parametrizing point-line (27) We represent the Cartan subalgebra in our root diagram by two bullets at the origin.…”

Conjugates of strongly equivariant maps

“…These are defined in [10], where Kuga calls r a (generalized) Eichler map. For the second application, to compactification of arithmetic varieties, see [20] and also [4], where strongly equivariant maps are called symmetric maps.…”