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The Kähler geometry of toric manifolds

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 Kähler 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 Kähler 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.