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K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. (English) Zbl 1353.14051
Let \(X\) be an \(n\)-dimensional Fano manifold, i.e., a compact complex manifold with ample anticanonical bundle \(K_X^{-1}\). A Kähler-Einstein metric on \(X\) is a Kähler metric \(\omega\) such that \[ \mathrm{Ric}(\omega)=\omega. \] This implies that \([\omega]=c_1(K_X^{-1})\), so conversely to construct a Kähler-Einstein metric, one starts with a Kähler metric \(\omega_0\) cohomologous to \(c_1(K_X^{-1})\) and seeks a smooth function \(\varphi\) such that \(\omega=\omega_0+\sqrt{-1}\partial\overline{\partial}\varphi\) is a Kähler-Einstein metric. This translates into a complex Monge-Ampère equation for \(\varphi\), which is not always solvable. A famous conjecture of Yau-Tian-Donaldson predicts that a Kähler-Einstein metric exists if and only if \((X,K_X^{-1})\) is K-polystable (an algebro-geometric notion of stability).
S.K. Donaldson [J. Differ. Geom. 70, No. 3, 453–472 (2005; Zbl 1149.53042)] proved that existence of Kähler-Einstein metrics implies K-semistability, and this was strengthened to K-polystability by J. Stoppa [Adv. Math. 221, No. 4, 1397–1408 (2009; Zbl 1181.53060)] assuming that the automorphism group of \(X\) is discrete, and to relative K-polystability by J. Stoppa and G. Székelyhidi [J. Eur. Math. Soc. (JEMS) 13, No. 4, 899–909 (2011; Zbl 1230.53069)].
The main result of this paper is to finally show that existence of Kähler-Einstein metrics implies K-polystability in general. The author also extends this to mildly singular \(\mathbb{Q}\)-Fano varieties, and to the logarithmic setting.

MSC:
14J45 Fano varieties
32Q20 Kähler-Einstein manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Software:
PALP
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