On the Relative Twist Formula of l-adic Sheaves

We propose a conjecture on the relative twist formula of l-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.