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The aim of the mini-course is to give an overview of recent results about the existence of Kähler-Einstein (KE) metrics on Fano varieties and to explain how such
canonical metrics are related to the construction of "nice and natural" compact moduli spaces of certain stable Fano varieties.
We will start by a discussion of classical results about the problem of existence (and obstruction to) of KE metrics on Fano manifolds, introducing the "tools" used in the subject. Next we will be focused on the algebricity of Gromov-Hausdorff limits of polarized Kähler manifolds under non-collapsing hypothesis and its consequences: such limits are of fundamental importance both in the proof of the equivalence between existence of KE metrics and K-stability (the so-called "Yau-Tian-Donaldson" conjecture in the Fano case) and in the construction of canonical compactifications of moduli of KE/K-stable manifolds. Finally, I will address the problem of existence of singular metrics for smoothable K-stable Fano varieties and their role in the moduli compactification, together with a discussion of the "explicit" two dimensional Del Pezzo case and of natural open problems. SHORT BIBLIOGRAPHY for the mini-course: (1) For an introduction to the subject: -G. Tian, "Canonical metrics in Kahler Geometry". Lectures in Mathematics, ETH Zurich, 2000. (2) For GH limits: -J. Cheeger, "Degeneration of Riemannian metrics under Ricci curvature bounds", Lezioni Fermiane, Pisa 2001. -S. Donaldson, S. Sun. "Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry" Acta Math. 213 (2014), no.1, 63-106. (3) For K-stability implies KE: -X-X. Chen, S. Donaldson, S. Sun. "Kahler-Einstein metrics on Fano manifolds, I-II-III" J. Amer. Math. Soc. 28 (2015), (4) For singular KE metrics (with emphasis in the Fano case): -R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, "Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties", arXiv:1111.7158 (5) For KE\K-moduli spaces and existence of KE metrics on singular smoothable Fano varieties: -Y. Odaka, C. Spotti, S. Sun, "Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics", J. Differential Geom., Volume 102, Number 1 (2016), 127-172. -C. Spotti, S. Sun, C. Yao, "Existence and deformations of Kähler Einstein metrics on smoothable Q-Fano varieties", (2014), arXiv:1411.1725, to appear on Duke Math. J. -Y. Odaka: "Compact moduli spaces of Kahler-Einstein Fano varieties", (2014) Publications of the Research Institute for Mathematical Sciences. -C. Li, X. Wang, C. Xu, "Degeneration of Fano Kahler-Einstein manifolds", (2014) arXiv:1411.0761. |