TY - JOUR
T1 - On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
AU - Li, Chi
AU - Wang, Xiaowei
AU - Xu, Chenyang
N1 - Funding Information:
The first author was partially supported by National Science Foundation (NSF) grant DMS-1405936. The second author was partially supported by Simons Foundation Collaboration Grant for Mathematicians 281299 and by NSF grant DMS-1609335. The third author was partially supported by a Recruitment Program of Foreign Experts grant; his visits to the Institute for Advanced Study were partially supported by Ky Fan and Yu-Fen Fan Membership Funds, the S. S. Chern Foundation, and NSF grants DMS-1128155 and DMS-1252158.
Funding Information:
We are very grateful to Jacob Sturm formany valuable suggestions and comments.We would also like to thank Jarod Alper, Daniel Greb, Reyer Sjamaar, and Chris Woodward for helpful comments, as well as the anonymous referees for numerous useful suggestions, and most particularly the authors of [47] for bringing their work to our attention. The first author thanks D. H. Phong, Jacob Sturm, and Jian Song for their constant encouragement over the years. A large part of this work was done during the third author's stay at the Institute for Advanced Study in Princeton, and he thanks that institution for its hospitality. The first author was partially supported by National Science Foundation (NSF) grant DMS-1405936. The second author was partially supported by Simons Foundation Collaboration Grant for Mathematicians 281299 and by NSF grant DMS-1609335. The third author was partially supported by a Recruitment Program of Foreign Experts grant; his visits to the Institute for Advanced Study were partially supported by Ky Fan and Yu-Fen Fan Membership Funds, the S. S. Chern Foundation, and NSF grants DMS-1128155 and DMS-1252158.
Publisher Copyright:
© 2019, Duke University Press.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - 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.
AB - 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.
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U2 - 10.1215/00127094-2018-0069
DO - 10.1215/00127094-2018-0069
M3 - Article
AN - SCOPUS:85067118479
SN - 0012-7094
VL - 168
SP - 1387
EP - 1459
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 8
ER -