We describe a
-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
-adic,
-adic,
and Betti cohomologies, 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.
