Rigid Local Systems and Weighted Homogeneous Curves

TL;DR

This paper introduces a new concept of rigid local systems on the complements of weighted homogeneous plane curves, linking it to classical rigidity and constructing large families with special multivalued functions.

Contribution

It defines a novel notion of rigidity for local systems on plane curve complements and connects it to classical concepts, providing explicit constructions.

Findings

Established a new rigidity notion for local systems on plane curve complements.

Linked the new notion to classical rigidity on the Riemann sphere.

Constructed large families of rigid local systems with special multivalued functions.

Abstract

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.