A now classical construction due to Kato and Nakayama attaches
a topological space (the “Betti realization”) to a log scheme over
. We
show that in the case of a log smooth degeneration over the standard log disc, this
construction allows one to recover the topology of the germ of the family from the log
special fiber alone. We go on to give combinatorial formulas for the monodromy and
the
differentials acting on the nearby cycle complex in terms of the log structures. We
also provide variants of these results for the Kummer étale topology. In the case of
curves, these data are essentially equivalent to those encoded by the dual graph of a
semistable degeneration, including the monodromy pairing and the Picard–Lefschetz
formula.
