  We study the vanishing cycle complex $\varphi_fA_X$ for a holomorphic
function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain
(for instance, a localization of the ring of integers of a cyclotomic field,
where the monodromy eigenvalue decomposition may hold after a localization of
$A$). Assuming the perversity of the shifted constant sheaf $A_X[d_X]$, we show
that the lowest possibly-non-zero vanishing cohomology at $0\in X$ can be
calculated by the restriction of $\varphi_fA_X$ to an appropriate nearby curve
in the singular locus $Y$ of $f$, which is given by intersecting $Y$ with the
intersection of sufficiently general hyperplanes in the ambient space passing
sufficiently near 0. The proof uses a Lefschetz type theorem for local
fundamental groups. In the homogeneous polynomial case, a similar assertion
follows from Artin's vanishing theorem. By a related argument we can show the
vanishing of the non-unipotent monodromy part of the first Milnor cohomology
for many central hyperplane arrangements with ambient dimension at least 4.
