Foliated and Mather-Jacobian discrepancies via tangential arcs

Summary · qwen2.5:32b

The article introduces a tangential arc-space approach to calculate foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, focusing on an invariant normal crossing separatrix divisor. This method links foliated adjunction with ordinary log pairs and applies the Ein-Mustață-Yasuda arc-space theorem to derive a tangential codimension formula, aligning logarithmic codimensions of toroidal tangential divisorial cylinders with their discrepancies. The theory supports a branch-conductor description for identifying non-lc and non-klt loci in foliated settings.

Excerpt

arXiv:2607.01809v2 Announce Type: replace Abstract: This article develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, relative to a fixed invariant normal crossing separatrix divisor. In the non-resonant logarithmic case, reduced tangential arcs centred on the prescribed tangential locus are confined to this divisor. The tangential sector is therefore presented, at the reduced arc level, by the normalised separatrix-conductor system. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs obtained by adjunction on the normalised branches and conductors. The arc-space theorem of Ein-Musta\c{t}\u{a}-Yasuda, applied on these strata, then gives a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. For toroidal invariant divisors read on branches, this tangential discrepancy agrees with the usual foliated discrepancy. The resulting theory gives toroidal tangential inversion of adjunction, a branch--conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather-Jacobian refinement for the canonical image separatrix system.