Abstract
We consider the coherent cohomology of toroidal compactifications of locally symmetric varieties (such as Shimura varieties) with coefficients in the canonical and subcanonical extensions of automorphic vector bundles, and give explicit conditions for them to vanish in certain degrees. We also provide algorithms for determining all such degrees in practice.
Highlights
Background materials2.1 Locally symmetric varietiesLet G be a reductive algebraic group over Q such that G(R) acts transitively on H, a finite disjoint union of Hermitian symmetric domains
Let h0 be a fixed choice of a point of H, so that H = G(R)h0, and let H0 denote the connected component of h0, which is a Hermitian symmetric domain by assumption
The justification for this is that the geometric structures of the resulted (V [μ], ∇) and their canonical and subcanonical extensions only depend on the weights μ in [μ], but not on the structure of V[μ] as a representation of GC
Summary
Let G be a reductive algebraic group over Q such that G(R) acts transitively on H, a finite disjoint union of Hermitian symmetric domains. Let h0 be a fixed choice of a point of H, so that H = G(R)h0, and let H0 denote the connected component of h0, which is a Hermitian symmetric domain by assumption. Suppose H0 ∼= G0(R)/K0 for some maximal compact subgroup K0 of G0(R), which can be identified with the stabilizer of h0 in G0(R). Suppose X is a complex-analytic manifold such that there exist finitely many neat arithmetic subgroups i of G(Q) stabilizing H0 and gi ∈ G(R) such that X ∼= i (gi igi−1)\(giH0) ∼= i( i\H0). By [3], X has the structure of a (possibly disconnected) quasi-projective variety, embedded in its minimal compactification Xmin ∼= i ( i\H0)min, the latter being a projective normal variety. X)red (with its reduced structure) is a simple normal crossings divisor, which is equipped with a canonical proper surjective morphism : Xtor → Xmin
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