Cohomological Milnor formula and Saito's conjecture on characteristic classes
Enlin Yang, Yigeng Zhao
TL;DR
The paper proves a quasi-projective case of Saito's conjecture by constructing cohomological characteristic classes $C_{X/S}(\mathcal{F})$ and a new non-acyclicity class $C^{Z}_{X/Y/S}(\mathcal{F})$, relating them via a fibration formula that splits the relative class into a Milnor-type component and an acyclicity correction. A generalized transversality framework is developed to handle base-change and localization, enabling a robust six-functor compatible theory in relative settings. The authors derive a cohomological Milnor formula on curves, a Grothendieck–Ogg–Shafarevich type formula, and a cohomological conductor formula, and deduce the quasi-projective case of Saito's conjecture by induction on dimension using a fibration strategy. The work connects local acyclicity, vanishing cycles, and characteristic cycles through an array of base-change, specialization, and push-forward results, and leverages ∞-categorical tools to manage the lifting and coherence aspects of the constructions. Overall, the results provide a powerful cohomological framework that expresses ramification invariants via characteristic cycles and relative trace maps, with potential applications to motivic six-functor theories and mixed-characteristic conjectures.
Abstract
We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic class supported on the non-acyclicity locus of a separated morphism relatively to a constructible sheaf. As applications of the functorial properties of this class, we prove cohomological analogs of the Milnor formula and the conductor formula for constructible sheaves on (not necessarily smooth) varieties.