The Meyer functions for projective varieties and their application to local signatures for fibered 4–manifolds

Abstract

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 has 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 nonhyperelliptic fibrations of genus 4 and 5 and compute some examples.

Meyer functions and the signature of fibered 4-manifolds

“…Let g be the genus of the fibers and let ρ X : π 1 (U X ) → M g be the monodromy of this family. Theorem 4.5 (Kuno [26]). There exists a unique Q-valued 1-cochain φ X : π 1 (U X ) → Q whose coboundary equals the pull-back ρ * X τ g .…”

“…Similar constructions are possible for generic non-hyperelliptic fibrations of genus 4 and 5. For details, see [26].…”

A local signature for fibered 4-manifolds with a finite group action

“…Local signatures for many kinds of restricted classes of fibrations are listed in Ashikaga-Endo [1] and Ashikaga-Konno [2]. Recently, Iida [12], Kuno [14] [15] and Yoshikawa [25] constructed it for some restricted classes.…”