Existence and deformations of Kähler-Einstein metrics on smoothable \(\mathbb{Q}\)-Fano varieties. (English) Zbl 1362.53082

Spotti, Cristiano; Sun, Song; Yao, Chengjian

Duke Math. J. 165, No. 16, 3043-3083 (2016).

Motivated by the study of the moduli space of K-polystable Fano manifolds, the authors here consider what happens at the “boundary”, i.e., whether \(\mathbb{Q}\)-Gorenstein smoothable \(\mathbb{Q}\)-Fano varieties admit weak Kähler-Einstein metrics (i.e., Kähler currents with continuous local potentials that are smooth KE metrics on the regular part).
Indeed, the main theorem of the article (Theorem 1.1) states that such a Gorenstein smoothable K-polystable variety \(X_0\) admits a weak KE metric \(\omega_0\). Moreover, assuming that the automorphism group of \(X_0\) is discrete, if we take a smoothing of \(X_0\) by smooth varieties \(X_t\), then \(X_t\) admits a smooth KE metric \(\omega_t\) for all small \(t\) such that the Gromov-Hausdorff limit of \(\omega_t\) is \(\omega_0\).
The idea of the proof is Donaldson’s continuity method of deforming the cone angle of a conical Kähler-Einstein metric. The technical difficulties lie in the fact that weak conical KE metrics behave in an unknown manner near the singularities of the variety. This is circumvented by approximating them by corresponding objects on nearby smooth varieties.

Reviewer: Vamsi Pritham Pingali (Bangalore)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
32J99	Compact analytic spaces
14J10	Families, moduli, classification: algebraic theory
14J45	Fano varieties
32Q20	Kähler-Einstein manifolds

Keywords:

Kähler-Einstein; K-stability; Fano; moduli; separatedness; compactification; Monge-Ampère; Gromov-Hausdorff