  We prove that in rank 1, the Abbes-Saito log conductor is determined by its
restriction to curves. This result is essentially established by analyzing
Artin-Schreier-Witt extensions. Consequently, we confirm an expectation of H.
Esnault and M. Kerz. We also conjecture that this result holds in arbitrary
finite rank. As an application, we translate recent results in higher class
field theory of M. Kerz and S. Saito to the log-ramification context.
