  We describe a 3-step filtration on all logarithmic abelian varieties with
constant degeneration. The obstruction to descending this filtration, as a
variegated extension, from logarithmic geometry to algebraic geometry is
encoded in a bilinear pairing valued in the characteristic monoid of the base.
This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti
cohomolgies, and recovers the Picard-Lefschetz transformation in the case of
Jacobians. The Hodge realization of the filtration is the monodromy weight
filtration on the limit mixed Hodge structure.
