  We prove a relative version of the Picard-Lefschetz theorem, describing the
variation of relative homology groups $H_d(Y_t \setminus A_t,B_t\setminus A_t)$
in the fibers of a smooth fiber bundle $Y \to T$ of complex manifolds with
$A\cup B \subset Y$ transverse. From this we derive the vanishing of certain
iterated variations, a system of constraints dubbed "hierarchy". As
applications, we rederive the known analytic structure of Aomoto polylogarithms
and massive one loop Feynman integrals. Moreover, we introduce the "simple
type" to prove hierarchy constraints in degenerate cases where the
Picard-Lefschetz formula does not apply, e.g. the massless triangle or the ice
cream cone Feynman diagram. We compare our findings with a "classical"
hierarchy of iterated variations (from 1960's $S$-matrix theory) and show how
our setup not only explains, but also refines the latter. In order to do so, we
need to further resolve the geometry of Feynman motives: We boldly blow up what
no one has blown up before.
