The connection between Kähler-Einstein metrics on Fano manifolds and stability notions in algebraic geometry, often referred to as the Yau-Tian-Donaldson conjecture, is a topic in Kähler geometry that has been actively investigated in the last few decades and many deep results are known. This lecture series focuses on its finite dimensional approximations, namely the correspondence between the anticanonically balanced metrics on Fano manifolds and the (finite dimensional) stability in the sense of Geometric Invariant Theory.

The first lecture concerns the basics of anticanonically balanced metrics and the variational setup, following [1]. We briefly touch on Kähler-Einstein metrics (particularly [2]) and discuss how anticanonically balanced metrics approximate Kähler-Einstein metrics, following Berman-Witt Nyström [3].

The second lecture concerns the algebro-geometric side, with a particular focus on the Ding stability (see e.g. [4]) and the stability introduced by Saito-Takahashi [10], which can be regarded as the Hilbert-Mumford criterion for the stability conditions for varieties. We also discuss the slope formula for various energy functionals established by many people; the reference for this topic is [5].

The third lecture proves the main result in [7], which states that the existence of anticanonically balanced metrics is equivalent to the stability of Saito-Takahashi. We finally discuss how we can extend the aforementioned results to coupled Kähler-Einstein metrics introduced by Hultgren-Witt Nyström [8] (more precisely the coupled anticanonically balanced metrics defined by Takahashi [11]), which involves a modification of the coupled K-stability defined in [8]. Time permitting, we comment on the relationship to the results in [9] which relate anticanonically balanced metrics to the delta_m-invariant introduced by Fujita-Odaka [6].

Prerequisite:

Basic knowledge of complex differential geometry and Kähler manifolds. Familiarity with the basics of complex algebraic geometry is desirable but not essential.

References:

[1] R. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Ampère equations. Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179--245. MR3090260

[2] 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. MR4334189

[3] R. Berman and D. W. Nyström, Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons, arXiv preprint arXiv:1401.8264

[4] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841. MR3669511

[5] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability and asymptotics of energy functionals in Kähler geometry, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 9, 2905–2944. MR3985614

[6] K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521. MR3896135

[7] Y. Hashimoto, Anticanonically balanced metrics and the Hilbert-Mumford criterion for the \delta_m-invariant of Fujita-Odaka, arXiv:2104.12346 (2021).

[8] J. Hultgren and D. W. Nyström, Coupled Kähler-Einstein metrics, Int. Math. Res. Not. IMRN 21 (2019), 6765– 6796. MR4027565

[9] Y. A. Rubinstein, G. Tian, and K. Zhang, Basis divisors and balanced metrics, arXiv preprint arXiv:2008.08829, to appear in J. Reine Angew. Math. (2020).

[10] S. Saito and R. Takahashi, Stability of anti-canonically balanced metrics, Asian J. Math. 23 (2019), no. 6, 1041-1058. MR4136489

[11] R. Takahashi, Geometric quantization of coupled Kähler-Einstein metrics. Anal. PDE 14 (2021), no. 6, 1817--1849. MR4308666

Notes download:

Talk1.pdf Talk2.pdf Talk3.pdf