The Grothendieck-Katz Conjecture for privileged local systems

We define study privileged local systems on a smooth quasi-projective complex variety $X$ with fixed quasi-unipotent monodromies around a boundary divisor. The notion is a generalization of Katz’ physical rigidity on an open of ${\mathbb{P}}^1$. It is defined in any dimension and includes non-rigid local systems. In the latter case, Katz used the \emph{middle convolution} procedure to classify all physically local systems and derived several consequences such as the $p$-curvature conjecture for local systems of this type. We present a new proof, which avoids the use of middle convolution. This yields examples in higher dimension of local systems which verify the $p$-curvature conjecture.