On the relative twist formula of $\mathrm{\ell }$-adic sheaves

We propose a conjecture on the relative twist formula of $\mathrm{\ell }$-adic sheaves, which can be viewed as a generalization of Kato-Saito’s conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.