  These lecture notes are written for a PhD mini-course I gave at the CIRM in
Luminy in 2019. Their intended purpose was to present, in the context of smooth
toric varieties, a relatively self-contained and elementary introduction to the
theory of extremal K\"ahler metrics pioneered by E. Calabi in the 1980's and
extensively developed in recent years. The framework of toric manifolds, used
in both symplectic and algebraic geometry, offers a fertile testing ground for
the general theory of extremal K\"ahler metrics and provides an important class
of smooth complex varieties for which the existence theory is now understood in
terms of a stability condition of the corresponding Delzant polytope. The notes
do not contain any original material nor do they take into account some more
recent developments, such as the non-Archimedean approach to the Calabi
problem. I am making them available on the arXiv because I continue to get
questions about how they can be cited.
