  We study a secondary invariant, called the Meyer function, on the fundamental
group of the complement of the dual variety of a smooth projective variety.
This invariant have played an important role when studying the local signatures
of fibered 4-manifolds from topological point of view. As an application of our
study, we define a local signature for generic non-hyperelliptic fibrations of
genus 4 and 5 and compute some examples.
