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.
"Resolution of the canonical fiber metrics for a Lefschetz fibration." J. Differential Geom. 108 (2) 295 - 317, February 2018. https://doi.org/10.4310/jdg/1518490819