  We study Deligne's conjecture on the monodromy weight filtration on the
nearby cycles in the mixed characteristic case, and reduce it to the
nondegeneracy of certain pairings in the semistable case. We also prove a
related conjecture of Rapoport and Zink which uses only the image of the Cech
restriction morphism, if Deligne's conjecture holds for a general hyperplane
section. In general we show that Deligne's conjecture is true if the standard
conjectures hold.
