The minimal exponent and k-rationality for local complete intersections

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 α ˜(Z)>k+r, where α ˜(Z) is the minimal exponent of Z. We also characterize this condition in terms of the Hodge filtration on the intersection complex Hodge module of Z. Furthermore, we show that if Z has k-rational singularities, then the Hodge filtration on the local cohomology sheaf ℋ Z r (𝒪 X ) is generated at level dim(X)-⌈α ˜(Z)⌉-1 and, assuming that k≥1 and Z is singular, of dimension d, that ℋ k (Ω ̲ Z d-k )≠0. All these results have been known for hypersurfaces in smooth varieties.