The aim of this lecture series is to present some results on perturbation problems for extremal Kähler metrics, which are a generalisation of constant scalar curvature Kähler (cscK) metrics. The goal in these problems is to create new such metrics from old ones, via analytic techniques where the linearisation of the equation plays a central role. The first lecture will be on background on cscK and extremal metrics (following [13]). To prepare for the perturbation problems I will discuss, a particular emphasis will be on the linear theory, where automorphisms play a key role. I will then highlight the strategy and method of proofs in these problems by going through the proof of a classical such theorem -- the LeBrun-Simanca openness theorem ([10]). I will also discuss the statements of some other perturbation problems (without proofs). In the second lecture, I will discuss the results of joint work with Spotti ([11]). This falls under the general theme of constructing extremal metrics on the total space of fibrations, which has a rich history of study ([9, 8, 4, 5, 7]). I will explain how the result fits into this story, and the new elements. In the third and final lecture, I will discuss recent joint work with Dervan ([6]). This result concerns the construction of extremal metrics on blowups. Again, this is a problem with a long history of study ([1, 2, 3, 12, 14]), which have given sufficient conditions for the blowup of an extremal manifold to admit extremal metrics. We also are able to deal with blowing up certain semistable manifolds, a case which has not been considered before. I will explain how the result compares to previous constructions, and the novelty of our approach.

Some basic knowledge of differential geometry, complex manifolds and Kähler geometry.

[2] -------, Blowing up Kähler manifolds with constant scalar curvature. II, Ann. of Math. (2) 170 (2009), no. 2, 685-738. MR 2552105 (2010m:32025)

[3] Claudio Arezzo, Frank Pacard, and Michael Singer, Extremal metrics on blowups, Duke Math. J. 157 (2011), no. 1, 1-51. MR 2783927 (2012k:32024)

[4] Till Brönnle, Extremal Kähler metrics on projectivized vector bundles, Duke Math. J. 164(2015), no. 2, 195-233. MR 3306554

[5]  Ruadhaí Dervan and Lars Martin Sektnan, Extremal metrics of fibrations, Proc. Lond. Math. Soc. (3) 120 (2020), no. 4, 587-616. MR 4008378

[6] -------, Extremal Kähler metrics on blowups, (2021), arXiv:2110.13579.

[7] -------, Optimal symplectic connections on holomorphic submersions, Comm. Pure Appl. Math. 74 (2021), no. 10, 2132-2184. MR 4303016

[8] Joel Fine, Constant scalar curvature Kähler metrics on  bred complex surfaces, J. Differential Geom. 68 (2004), no. 3, 397-432. MR 2144537

[9] Ying-Ji Hong, Constant Hermitian scalar curvature equations on ruled manifolds, J. Differential Geom. 53 (1999), no. 3, 465-516. MR 1806068

[10] Claude LeBrun and Santiago R. Simanca, On the Kähler classes of extremal metrics, Geometry and global analysis (Sendai, 1993), Tohoku Univ., Sendai, 1993, pp. 255-271. MR 1361191

[11] Lars Martin Sektnan and Cristiano Spotti, Extremal metrics on the total space of destabilising test configurations, (2021), arXiv:2110.07496.

[12] Gábor Székelyhidi, On blowing up extremal Kähler manifolds, Duke Math. J. 161 (2012), no. 8, 1411-1453. MR 2931272

[13] -------, An introduction to extremal Kähler metrics, Graduate Studies in Mathematics, vol.152, American Mathematical Society, Providence, RI, 2014. MR 3186384

[14] -------, Blowing up extremal Kähler manifolds II, Invent. Math. 200 (2015), no. 3, 925-977. MR 3348141

Notes download：

Note: Notes.pdf

Talk1: Talk 1.pdf

Talk2: Talk 2.pdf

Talk3: Talk 3.pdf