On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties

Abstract

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano Kähler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space.

41 pages. Final version. Minor change with exposition improved. To appear in Duke Math. J

References in corpus (3)

• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• Lectures on Stability and Constant Scalar Curvature
• Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties

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• Applications of the moduli continuity method to log K-stable pairs
• On K-stability of some del Pezzo surfaces of Fano index 2
• Moduli theory, stability of fibrations and optimal symplectic connections
• Reductive quotients of klt singularities
• Kähler-Einstein metrics on families of Fano varieties
• Foldable fans, cscK surfaces and local K-moduli
• On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves
• Weighted cscK metrics on Kähler varieties