We develop a theory of multi-stage degenerations of toric varieties over
finite rank valuation rings, extending the Mumford--Gubler theory in rank one.
Such degenerations are constructed from fan-like structures over totally
ordered abelian groups of finite rank. Our main theorem describes the geometry
of successive special fibers in the degeneration in terms of the polyhedral
geometry of a system of recession complexes associated to the fan.Comment: 13 pages. v3: Added Example 4.1.8 and new references. To appear in
Bulletin of the London Mathematical Societ
