The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails

2023/12/20 by Baudin, Jefferson, Bernasconi, Fabio, Kawakami, Tatsuro · 7 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.13456

Abstract

We show that the Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails for threefolds in any positive characteristic, and for terminal 3-folds in characteristic p ∈ \2, 3, 5\. To prove this, we introduce the notion of \mathbbF_p-rationality for singularities in positive characteristic and we show that klt singularities in dimension at most 4 are \mathbbF_p-rational. We apply this to prove a Frobenius--stable version of the Kawamata--Viehweg vanishing theorem on K-trivial 3-folds.

Cited by

• Effective characterization of ordinary abelian varieties, and beyond
• On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension
• Generic vanishing theory in positive characteristic
• On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type
• On Gorenstein ℚ_p-rational threefold and fourfold singularities
• A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves
• Duality between W_n-Cartier crystals and ℤ/pⁿℤ-perverse sheaves

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