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Abstract We prove Grauert–Riemenschneider–type vanishing theorems for excellent divisiorally log terminal threefolds pairs whose closed points have perfect residue fields of positive characteristic $p>5$. Then we discuss applications to dlt singularities and to Mori fiber spaces of three-folds.
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An analogue of adjoint ideals and PLT singularities in mixed characteristic
We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely F F -regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic p > 5 p>5 , which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5 p>5 . In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5 p > 5 . Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze.
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- PAR ID:
- 10326771
- Date Published:
- Journal Name:
- Journal of Algebraic Geometry
- Volume:
- 31
- Issue:
- 3
- ISSN:
- 1056-3911
- Page Range / eLocation ID:
- 497 to 559
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1090/jag/797
