Semi-continuity for conductor divisors of étale sheaves
Haoyu Hu, Jean-Baptiste Teyssier
TL;DR
This work extends semi-continuity results for ramification conductors from relative curves to higher relative dimensions in a geometric setting. It introduces generic conductor divisors $\mathrm{GC}_{f}(j_!\mathscr F)$ and generic logarithmic conductor divisors $\mathrm{GLC}_{f}(j_!\mathscr F)$ for étale sheaves on a complement $U=X\setminus D$, and proves that there exist dense open subsets of the base over which these generic divisors pull back to the actual fibre-wise conductor divisors, while over the complement they satisfy lower bounds. The argument leverages Abbes–Saito ramification theory, curve-cutting arguments, and base-change techniques to establish lower semi-continuity of both the total and logarithmic conductor divisors in families, with parallel results for the corresponding conductor invariants $\mathrm{c}_K(M)$ and $\mathrm{lc}_K(M)$. The results form an $\ell$-adic analogue of the Poincaré–Katz irregularity semi-continuity in the D-module setting and unify previous curve-relative results with higher-dimensional ramification theory, enabling new Betti-number estimates in families of perverse sheaves. The work thus provides a robust framework for understanding how ramification data varies in families of étale sheaves in higher dimensions.
Abstract
In this article, we prove a semi-continuity property for both conductor divisors and logarithmic conductor divisors for étale sheaves on higher relative dimensions in a geometric situation. It generalizes a semi-continuity result for conductors of étale sheaves on relative curves to higher relative dimensions, and it can be considered as a higher dimensional $\ell$-adic analogy of André's result on the semi-continuity of Poincaré-Katz ranks of meromorphic connections on smooth relative curves.