Duke Mathematical Journal

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

Chi Li, Xiaowei Wang, and Chenyang Xu

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

Article information

Duke Math. J., Volume 168, Number 8 (2019), 1387-1459.

Received: 7 August 2017
Revised: 26 November 2018
First available in Project Euclid: 3 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J45: Fano varieties
Secondary: 14J10: Families, moduli, classification: algebraic theory 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

K-stability Kähler–Einstein metrics Q-Fano varieties geometric invariant theory Gromov–Hausdorff limit Artin stack good moduli space


Li, Chi; Wang, Xiaowei; Xu, Chenyang. On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties. Duke Math. J. 168 (2019), no. 8, 1387--1459. doi:10.1215/00127094-2018-0069. https://projecteuclid.org/euclid.dmj/1556848995

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