On the proper moduli spaces of smoothable K\"ahler-Einstein Fano varieties
Algebraic Geometry
2019-05-29 v4 Differential Geometry
Abstract
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-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 K\"ahler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of -Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space.
Keywords
Cite
@article{arxiv.1411.0761,
title = {On the proper moduli spaces of smoothable K\"ahler-Einstein Fano varieties},
author = {Chi Li and Xiaowei Wang and Chenyang Xu},
journal= {arXiv preprint arXiv:1411.0761},
year = {2019}
}
Comments
41 pages. Final version. Minor change with exposition improved. To appear in Duke Math. J