The first lecture will be quick background on the constant scalar curvature Kähler (cscK) problem [1-5] and the general setup on cscK cone metrics mainly from the paper [6]. The second lecture will lead to the existence results for the cscK cone metrics, and more general, the singular cscK metrics [7]. After all these preparation, the third lecture will present results on geometric applications, including links to log K stabilities and properties of the cscK cone path etc [8].

[1] E. Calabi, Extremal Kähler metrics. Seminar on Differential Geometry, pp. 259–290, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982. MR0645743.

[2] E. Calabi, Extremal Kähler metrics. II. Differential geometry and complex analysis, 95–114, Springer, Berlin, 1985. MR0780039.

[3] X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics I,II, J. Amer. Math. Soc. 34 (2021), no. 4, 909-1009.

[4] S. K. Donaldson, Scalar curvature and projective embeddings. I. (English summary) J. Differential Geom. 59 (2001), no. 3, 479–522. MR1916953

[5] S. K. Donaldson, Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications, Springer, Heidelberg, 2012, pp. 49-79. MR2975584.

[6] K. Zheng, Geodesics in the space of Kähler cone metrics II: Uniqueness of constant scalar curvature Kähler cone metrics, Comm. Pure Appl. Math. 72 (2019), no. 12, 2621-2701. MR4020314.

[7] K. Zheng, Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability, arXiv:1803.09506.

[8] T. Aoi, Y. Hashimoto, K. Zheng, On uniform log K-stability for constant scalar curvature Kähler cone metrics, arXiv:2110.02518.

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