The aim of the course is to give an introduction to some topics of current interest in the global geometry of compact Kähler manifols.

**Part I** – Basic notions. Hermitian and Kähler manifolds, harmonic theory, Kodaira Vanishing and Embedding, Kempf-Ness Theorem, notions of canonical Kähler metrics.

Some textbooks:

- D. Huybrechts, Introduction to complex geometry, Springer.
- G. Székelyhidi, Introduction to extremal metrics, AMS.

**Part II** – K-stability and applications. Test configurations and Donaldson-Futaki invariant; Donaldson’s lower bound on the Calabi functional; Mabuchi functional and geodesics; sketch of the uniform, Fano case of the Yau-Tian-Donaldson Conjecture.

Some references:

- S. K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3.Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841.

**Part III** – Further topics (according to the interests of the audience). New equations in Kähler geometry, e.g. Z-critical connections and metrics; problems involving a deformation of the complex structure or a mirror map.

Some references:

- R. Dervan, J. B. McCarthy, L. M. Sektnan, Z-critical connections and Bridgeland stability conditions, arXiv.
- Scarpa, Carlo, Scalar curvature and deformations of complex structures. J. Reine Angew. Math. 797 (2023), 255–283.
- J. Stoppa, K-stability and large complex structure limits, arXiv.