The K\"ahler geometry of toric manifolds

TL;DR

This paper provides an accessible introduction to the theory of extremal K"ahler metrics on smooth toric varieties, emphasizing their geometric and algebraic properties and the role of stability conditions.

Contribution

It offers a self-contained overview of extremal K"ahler metrics in the context of toric manifolds, connecting symplectic, algebraic, and differential geometry.

Findings

Toric manifolds serve as a testing ground for extremal K"ahler metrics.

Existence of extremal metrics relates to stability of the Delzant polytope.

The notes clarify the theory without recent non-Archimedean developments.

Abstract

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.