Characteristic Epsilon Cycles of $\ell$-adic Sheaves on Varieties

Abstract

Let X be a smooth variety over a finite field F q . Let ℓ be a rational prime number invertible in F q . For an ℓ-adic sheaf F on X, we construct a cycle supported on the singular support of F whose coefficients are ℓ-adic numbers modulo roots of unity.It is a refinement of the characteristic cycle CC(F), in the sense that it satisfies a Milnor-type formula for local epsilon factors. After establishing fundamental results on the cycles, we prove a product formula of global epsilon factors modulo roots of unity. We also give a generalization of the results to varieties over general perfect fields.• For the ℓ-adic formalism of a noetherian topos T , we refer to [7], which we review in the appendix. The derived category of constructible complexes of E-sheaves (resp. O E -sheaves) on T is denoted by D b c (T, E) (resp. D b c (T, O E )). We put 0 for objects of D b c (T, O E ) (i.e. F 0 ∈ D b c (T, O E )) and denote F := F 0 ⊗ O E E.

Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic

“…For the total dimensions, this is achieved by the theory of characteristic cycles given by T. Saito [27]. For the local epsilon factors, epsilon cycles are defined in [30] as refinements of characteristic cycles. They treat the local epsilon factors of the vanishing cycles of ℓ-adic sheaves in a geometric way using cotangent bundles, but require that we ignore roots of unity, namely we treat the local epsilon factors in…”