  We present a novel strategy to systematically study complex-structure moduli
stabilization in Type IIB and F-theory flux compactifications. In particular,
we determine vacua in any asymptotic regime of the complex-structure moduli
space by exploiting powerful tools of asymptotic Hodge theory. In a leading
approximation the moduli dependence of the vacuum conditions are shown to be
polynomial with a dependence given by sl(2)-weights of the fluxes. This simple
algebraic dependence can be extracted in any asymptotic regime, even though in
nearly all asymptotic regimes essential exponential corrections have to be
present for consistency. We give a pedagogical introduction to the
sl(2)-approximation as well as a detailed step-by-step procedure for
constructing the corresponding Hodge star operator. To exemplify the
construction, we present a detailed analysis of several Calabi-Yau three- and
fourfold examples. For these examples we illustrate that the vacua in the
sl(2)-approximation match the vacua obtained with all polynomial and essential
exponential corrections rather well, and we determine the behaviour of the
tadpole contribution of the fluxes. Finally, we discuss the structure of vacuum
loci and their relations to several swampland conjectures. In particular, we
comment on the realization of the so-called linear scenario in view of the
tadpole conjecture.
