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We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.
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This content will become publicly available on December 1, 2025
Relative vanishing theorems for Q-schemes
We prove the relative Grauert–Riemenschneider vanishing, Kawamata–Viehweg vanishing, and Kollár injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses the Grothendieck limit theorem for sheaf cohomology and Zariski–Riemann spaces. We also show that these vanishing and injectivity theorems hold for locally Moishezon (respectively, projective) morphisms of quasi-excellent algebraic spaces and semianalytic germs of complex-analytic spaces (respectively, quasi-excellent formal schemes and non-Archimedean analytic spaces), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot’s theorem to all Noetherian Q-algebras by showing that pseudo-rationality descends under pure maps of Q-algebras. This solves a conjecture of Boutot and answers a question of Schoutens. The proofs of this Boutot-type result and of our vanishing and injectivity theorems all use a new characterization of rational singularities using Zariski–Riemann spaces.
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- PAR ID:
- 10585465
- Publisher / Repository:
- Foundation Compositio Mathematica
- Date Published:
- Journal Name:
- Algebraic Geometry
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2214-2584
- Page Range / eLocation ID:
- 84 to 144
- Subject(s) / Keyword(s):
- 14F17 14E15 14A15 14B15 13F40 14B05 vanishing theorems, excellent schemes, the Grothendieck limit theorem, Zariski–Riemann spaces, rational singularities
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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