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Linear invariance of intersections on unitary Rapoport–Zink spaces

  • Benjamin Howard EMAIL logo
Published/Copyright: June 14, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 5

Abstract

We prove an invariance property of intersections of Kudla–Rapoport divisors on a unitary Rapoport–Zink space.

Keywords: Rapoport–Zink spaces; Kudla–Rapoport conjecture
MSC 2010: 11G18; 14G35

Communicated by Jan Bruinier

Funding source: National Science Foundation

Award Identifier / Grant number: DMS1501583

Award Identifier / Grant number: DMS1801905

Funding statement: This research was supported in part by NSF grants DMS1501583 and DMS1801905.

A The exceptional divisor

Throughout this appendix we assume that 𝐤/ℚp is ramified. We want to explain why the Kudla–Rapoport divisors of Definition 2.4 are generally not flat over 𝒪˘𝐤.

Denote by 𝔽˘=𝒪˘𝐤/𝔪˘ the residue field of 𝒪𝐤˘. The two embeddings φ,φ¯:𝒪𝐤→𝒪˘𝐤 necessarily reduce to the unique ℤp-algebra morphism 𝒪𝐤→𝔽˘.

Definition A.1.

The exceptional divisorExc⊂M is the set of all points s∈M at which the action

i:𝒪𝐤→End⁢(Lie⁢(Xs))

is through scalars; that is to say, the action factors through the unique morphism 𝒪𝐤→𝔽˘. This is a closed subset of the underlying topological space of M, and we endow it with its induced structure of a reduced scheme over 𝔽˘.

Proposition A.2.

The exceptional divisor Exc⊂M is a Cartier divisor, and is isomorphic to a disjoint union of copies of the projective space Pn-1 over F˘.

Proof.

A point s∈M⁢(𝔽˘) corresponds to a pair (X0⁢s,Xs) over 𝔽˘, which we recall is really a tuple

(X0⁢s,i0,λ0,ϱ0,Xs,i,λ,FXs,ϱ)∈M⁢(𝔽˘).

If s∈Exc⁢(𝔽˘), then the action of 𝒪𝐤 on Lie⁢(X) is through the unique ℤp-algebra morphism 𝒪𝐤→𝔽˘. This implies that any codimension one subspace of F⊂Lie⁢(Xs) satisfies Krämer’s signature condition as in Section 2, and we obtain a closed immersion

ℙ⁢(Lie⁢(Xs)∨)↪Exc

by sending F↦(X0⁢s,i0,λ0,ϱ0,Xs,i,λ,F,ϱ). In other words, vary the codimension one subspace in Lie⁢(Xs) and leave all other data fixed.

It is clear that Exc is the disjoint union of all such closed subschemes, and that every connected component of Exc is reduced, irreducible, and of codimension one in M. The regularity of M then implies that Exc⊂M is defined locally by one equation. ∎

Proposition A.3.

Fix a nonzero x∈V, and any connected component D⊂Exc. For all k≫0 we have D⊂Z⁢(pk⁢x). In particular, Z⁢(pk⁢x) is not flat over O˘k.

Proof.

If we fix one point s∈D, Lemma 2.3 allows us to view

x∈Hom𝒪𝐤⁢(X0⁢s,Xs)⁢[1/p].

For all k≫0 we thus have pk⁢x∈Hom𝒪𝐤⁢(X0⁢s,Xs). It follows from the characterization of D≅ℙn-1 found in the proof of Proposition A.2 that the p-divisible groups X0⁢D and XD are constant (that is, are pullbacks via D→Spec⁢(𝔽˘) of p-divisible groups over 𝔽˘), and hence the restriction map

Hom𝒪𝐤⁢(X0⁢D,XD)→Hom𝒪𝐤⁢(X0⁢s,Xs)

is an isomorphism. Hence pk⁢x∈Hom𝒪𝐤⁢(X0⁢D,XD) and D⊂Z⁢(pk⁢x). ∎

References

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Received: 2019-01-25
Revised: 2019-05-14
Published Online: 2019-06-14
Published in Print: 2019-09-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Selberg integral over local fields
  3. On conjugation orbits of semisimple pairs in rank one
  4. On a topology property for the moduli space of Kapustin–Witten equations
  5. Differentiability of the evolution map and Mackey continuity
  6. A prime geodesic theorem for SL3(ℤ)
  7. Comparison and continuity of Wick-type star products on certain coadjoint orbits
  8. A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank
  9. Linear invariance of intersections on unitary Rapoport–Zink spaces
  10. Maximal subalgebras of finite-dimensional algebras
  11. A universal enveloping algebra for cocommutative rack bialgebras
  12. A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries
  13. Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver
https://doi.org/10.1515/forum-2019-0023
Keywords for this article
Rapoport–Zink spaces; Kudla–Rapoport conjecture

Articles in the same Issue

  1. Frontmatter
  2. Selberg integral over local fields
  3. On conjugation orbits of semisimple pairs in rank one
  4. On a topology property for the moduli space of Kapustin–Witten equations
  5. Differentiability of the evolution map and Mackey continuity
  6. A prime geodesic theorem for SL3(ℤ)
  7. Comparison and continuity of Wick-type star products on certain coadjoint orbits
  8. A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank
  9. Linear invariance of intersections on unitary Rapoport–Zink spaces
  10. Maximal subalgebras of finite-dimensional algebras
  11. A universal enveloping algebra for cocommutative rack bialgebras
  12. A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries
  13. Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver
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