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The space of shapes of a polyhedron with given total angles less than
at each
of its
vertices has a Kähler metric, locally isometric to complex hyperbolic space
. The
metric is not complete: collisions between vertices take place a finite distance from a
nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In
some interesting special cases, the metric completion is an orbifold. The concrete
description of these spaces of shapes gives information about the combinatorial
classification of triangulations of the sphere with no more than 6 triangles at a
vertex.
