Searching for canonical metrics on a Kahler manifold is an important topic in geometric analysis. The canonical metrics we will be focusing on are twisted Kahler-Einstein metrics, whose existence is equivalent to the solvability of certain Monge-Ampere type equations. An important problem is to find suitable algebro-geometric conditions that can guarantee the solvability of such equations. This kind of problem is often referred to as the Yau-Tian-Donaldson conjecture. In this lecture series we will present a quantization approach, which shows the existence of twisted Kahler-Einstein metrics on any polarized manifolds provided that certain algebraic invariant (called delta) is bigger than 1. This result removes the Fano assumption in the work of Berman-Boucksom-Jonsson.
The first lecture is largely a survey talk, together with all the basic notions needed for the proof of our main result.
The second lecture discusses the quantization techniques that will be crucially needed.
The third lecture will put all the ingredients together and produce the final proof.
Basic knowledge of complex differential geometry and Kähler manifolds. Familiarity with the basics of complex algebraic geometry is desirable but not essential.
 R. Berman, S. Boucksom, M. Jonsson, A variational approach to the Yau-Tian-Donaldson conjecture. J. Amer. Math. Soc. 34 (2021), no. 3, 605--652.
H. Blum and M. Jonsson. Thresholds, valuations, and K-stability. Adv. Math., 365:107062, 57, 2020.
Y.A. Rubinstein, G. Tian, K. Zhang, Basis divisors and balanced metrics, J. Reine Angew. Math. 778 (2021), 171–218.
K. Fujita and Y. Odaka. On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. (2), 70(4):511– 521, 2018.
K. Zhang, A quantization proof of the uniform Yau-Tian-Donaldson conjecture, arXiv:2102.02438, 2021.