Regular singular stratified bundles and tame ramification

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Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. $\mathcal {O}_X$-coherent $\mathscr {D}_{X/k}$-modules) on $X$ without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional $k$-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to $X$.