Resolution of the canonical fiber metrics for a Lefschetz fibration

TL;DR

This paper refines the understanding of fiber metrics in Lefschetz fibrations, showing that after a specific resolution, the family of constant curvature metrics behaves in a log-smooth manner near singular fibers, with detailed asymptotic structure.

Contribution

It demonstrates that the family of fiber metrics becomes log-smooth after an iterated blow-up resolution, extending previous results by Obitsu and Wolpert.

Findings

Family of fiber metrics is log-smooth after resolution.

Polyhomogeneous structure with integral powers and multiplicities.

Behavior described in terms of logarithmic parameters.

Abstract

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic.