The minimal exponent and k-rationality for local complete intersections

2022/12/04 by Qianyu Chen, Chen, Qianyu, Bradley Dirks +3 · 4 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2212.01898

Abstract

We show that if Z is a local complete intersection subvariety of a smooth complex variety X, of pure codimension r, then Z has k-rational singularities if and only if \widetildeα(Z)>k+r, where \widetildeα(Z) is the minimal exponent of Z. We also characterize this condition in terms of the Hodge filtration on the intersection cohomology Hodge module of Z. Furthermore, we show that if Z has k-rational singularities, then the Hodge filtration on the local cohomology sheaf H^r_Z(O_X) is generated at level dim(X)-\lceil \widetildeα(Z)\rceil-1 and, assuming that k≥ 1 and Z is singular, of dimension d, that H^k(\underlineΩ_Z^d-k)≠ 0. All these results have been known for hypersurfaces in smooth varieties.

Cited by

• Partial Cohomologically Complete Intersections via Hodge Theory
• Multiplier modules, V-filtrations and Bernstein-Sato polynomials on singular ambient varieties
• Inversion of adjunction for higher rational singularities
• Trace for the Du Bois complex

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