  Bridgeland stability manifolds of Calabi-Yau categories are of noticeable
interest both in mathematics and in physics. By looking at some of the known
example, a pattern clearly emerges and gives a fairly precise description of
how they look like. In particular, they all seem to have missing loci, which
tend to correspond to degenerate stability conditions vanishing on spherical
objects. Describing such missing strata is also interesting from a
mirror-symmetric perspective, as they conjecturally parametrize interesting
types of degenerations of complex structures. All the naive attempts at
constructing modular partial compactifications show how elusive and subtle the
problem in fact is: ideally, the missing strata would correspond to stability
manifolds of quotient triangulated categories, but establishing such
correspondence on geometric level and viewing stability conditions on quotients
of the original triangulated category as suitable degenerations of stability
conditions is not straightforward. In this paper, we will present a method to
construct such partial compactifications if some additional hypoteses are
satisfied, by realizing our space of interest as a suitable metric completion
of the stability manifold.

Algebraic Geometry