Characteristic cycles and the conductor of direct image

Abstract

We prove the functoriality for proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms to curves. The conductor formula in the constant coefficient case gives the geometric case of a formula conjectured by Bloch.

Trace and Künneth formulas for singularity categories and applications

“…In [Blo87], this formula is proven in relative dimension 1. Further results implying special cases of BCC have been obtained because then by Kato and Saito [KS05] and others, the most recent being a full proof in the geometric case by Saito [Sai17] (see § 5 for a more detailed discussion about the state of the art of BCC). In the mixed characteristic case, the conjecture is open in general outside the cases covered in [KS05].…”

“…2. When S is equicharacteristic, Conjecture 5.1.1 has been proved recently in [Sai17], based on Beilinson's theory of singular support of -adic sheaves. The special subcase of isolated singularities already appeared in [SGA7-I, Exp.…”

“…For higher relative dimensions, BCC is still open in its full generality, though there are many important partial positive results (see e.g. [KS05,Sai17]).…”

Characteristic class and the epsilon factor of an étale sheaf

“…1.11. Just after writing this paper, T. Saito [33] announced a proof for proper push-forward of characteristic cycles along projective morphisms f : X Ñ Y between smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support of F has dimension at most dim Y . Under the same assumption, T. Saito's result implies (1.9.1).…”