$$\ell $$-independence of the trace of local monodromy in a relative case (with an appendix by Qing Lu and Weizhe Zheng)

For a family of varieties, we prove that the alternating sum of the traces of “local” monodromy acting on the $$\ell $$ -adic etale cohomology groups of the generic fiber is an integer that is independent of $$\ell $$ . In the course of the proof, we also establish a result on fixed points.