  In the quests to uncovering the fundamental structures that underlie some of
the asymptotic Swampland conjectures we initiate the general study of
asymptotic period vectors of Calabi- Yau manifolds. Our strategy is to exploit
the constraints imposed by completeness, symmetry, and positivity, which are
formalized in asymptotic Hodge theory. We use these general principles to study
the periods near any boundary in complex structure moduli space and explain
that near most boundaries leading exponentially suppressed corrections must be
present for consistency. The only exception are period vectors near the
well-studied large complex structure point. Together with the classification of
possible boundaries, our procedure makes it possible to construct general
models for these asymptotic periods. The starting point for this construction
is the sl(2)-data classifying the boundary, which we use to construct the
asymptotic Hodge decomposition known as the nilpotent orbit. We then use the
latter to determine the asymptotic period vector. We explicitly carry out this
program for all possible one- and two-moduli boundaries in Calabi-Yau
threefolds and write down general models for their asymptotic periods.
