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 $\tilde{\alpha }(Z)>k+r$, where $\tilde{\alpha }(Z)$ 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 ${\mathcal{H}}_Z^r({\mathcal{O}}_X)$ is generated at level $\dim (X)-\lceil \tilde{\alpha }(Z)\rceil -1$ and, assuming that $k\ge 1$ and $Z$ is singular, of dimension $d$, that ${\mathcal{H}}^k({\underset{‾}{\mathrm{\Omega }}}_Z^{d-k})\mathrm{\ne }0$. All these results have been known for hypersurfaces in smooth varieties.