F-singularities via alterations

Abstract

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta )$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map ${\pi }_{\ast }{\omega }_Y\to {\omega }_X$ for every such alteration $\pi Y\to X$. Furthermore, building on work of B. Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-Viehweg vanishing theorems in the characteristic $p$ setting without assuming $W2$ lifting, and show that these are strong enough in some applications to extend sections.

38 pages, typos corrected, references updated and improved exposition. To appear in the American Journal of Mathematics

References in corpus (3)

• F-signature exists
• A survey of test ideals
• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems

Cited by in corpus (1)

• Irreducible Theta Divisors of PPAV's are Strongly F-regular