Rigid Local Systems and Weighted Homogeneous Curves

We introduce a notion of rigid local system on the comple- ment of a plane curve $Y$, which relies on a canonical Waldhausen de- composition of the Milnor sphere associated to $Y$. We show that when $Y$ is weigthed homogeneous this notion is deeply related to the classical notion of rigidity on the Riemann sphere. We construct large families of rigid local systems on the complement of weighted homogeneous plane curves and show that the corresponding $D$-modules are generated by `special' multivalued holomorphic functions.