One-dimensional components in the K-moduli of smooth Fano 3-folds
Authors:
Hamid Abban, Ivan Cheltsov, Elena Denisova, Erroxe Etxabarri-Alberdi, Anne-Sophie Kaloghiros, Dongchen Jiao, Jesus Martinez-Garcia and Theodoros Papazachariou
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/839
Published electronically:
October 9, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
References
- Hamid Abban and Ziquan Zhuang, K-stability of Fano varieties via admissible flags, Forum Math. Pi 10 (2022), Paper No. e15, 43. MR 4448177, DOI 10.1017/fmp.2022.11
- Jarod Alper, Harold Blum, Daniel Halpern-Leistner, and Chenyang Xu, Reductivity of the automorphism group of $K$-polystable Fano varieties, Invent. Math. 222 (2020), no. 3, 995–1032. MR 4169054, DOI 10.1007/s00222-020-00987-2
- Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süß, and Nivedita Viswanathan, The Calabi problem for Fano threefolds, London Mathematical Society Lecture Note Series, vol. 485, Cambridge University Press, Cambridge, 2023. MR 4590444
- Harold Blum, Daniel Halpern-Leistner, Yuchen Liu, and Chenyang Xu, On properness of K-moduli spaces and optimal degenerations of Fano varieties, Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 73, 39. MR 4292783, DOI 10.1007/s00029-021-00694-7
- Harold Blum, Yuchen Liu, and Chenyang Xu, Openness of K-semistability for Fano varieties, Duke Math. J. 171 (2022), no. 13, 2753–2797. MR 4505846, DOI 10.1215/00127094-2022-0054
- Harold Blum and Chenyang Xu, Uniqueness of K-polystable degenerations of Fano varieties, Ann. of Math. (2) 190 (2019), no. 2, 609–656. MR 3997130, DOI 10.4007/annals.2019.190.2.4
- David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
- Ivan Cheltsov, Igor Krylov, Jesus Martinez-Garcia, and Evgeny Shinder, On maximally non-factorial nodal Fano threefolds, Rev. Mat. Iberoam. 40 (2024), no. 5, 1781–1798. MR 4792548, DOI 10.4171/rmi/1478
- Ivan Cheltsov and Jihun Park, K-stable Fano threefolds of rank 2 and degree 30, Eur. J. Math. 8 (2022), no. 3, 834–852. MR 4498816, DOI 10.1007/s40879-022-00569-x
- Ivan Cheltsov and Hendrik Süß, K-polystability of two smooth Fano threefolds, Birational geometry, Kähler-Einstein metrics and degenerations, Springer Proc. Math. Stat., vol. 409, Springer, Cham, [2023] ©2023, pp. 165–193. MR 4606635, DOI 10.1007/978-3-031-17859-7_{8}
- Giulio Codogni and Zsolt Patakfalvi, Positivity of the CM line bundle for families of $K$-stable klt Fano varieties, Invent. Math. 223 (2021), no. 3, 811–894. MR 4213768, DOI 10.1007/s00222-020-00999-y
- D. F. Coray and M. A. Tsfasman, Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc. (3) 57 (1988), no. 1, 25–87. MR 940430, DOI 10.1112/plms/s3-57.1.25
- Elena Denisova, On $K$-stability of $\Bbb P^3$ blown up along the disjoint union of a twisted cubic curve and a line, J. Lond. Math. Soc. (2) 109 (2024), no. 5, Paper No. e12911, 26. MR 4745884, DOI 10.1112/jlms.12911
- Kento Fujita, On $K$-stability and the volume functions of $\Bbb {Q}$-Fano varieties, Proc. Lond. Math. Soc. (3) 113 (2016), no. 5, 541–582. MR 3570238, DOI 10.1112/plms/pdw037
- Kento Fujita, K-stability of Fano manifolds with not small alpha invariants, J. Inst. Math. Jussieu 18 (2019), no. 3, 519–530. MR 3936640, DOI 10.1017/s1474748017000111
- Kento Fujita, Uniform K-stability and plt blowups of log Fano pairs, Kyoto J. Math. 59 (2019), no. 2, 399–418. MR 3960299, DOI 10.1215/21562261-2019-0012
- Kento Fujita, A valuative criterion for uniform K-stability of $\Bbb Q$-Fano varieties, J. Reine Angew. Math. 751 (2019), 309–338. MR 3956698, DOI 10.1515/crelle-2016-0055
- Kento Fujita, On K-stability for Fano threefolds of rank 3 and degree 28, Int. Math. Res. Not. IMRN 15 (2023), 12601–12784. MR 4621852, DOI 10.1093/imrn/rnac190
- Priska Jahnke and Ivo Radloff, Terminal Fano threefolds and their smoothings, Math. Z. 269 (2011), no. 3-4, 1129–1136. MR 2860279, DOI 10.1007/s00209-010-0780-8
- Anne-Sophie Kaloghiros and Andrea Petracci, On toric geometry and K-stability of Fano varieties, Trans. Amer. Math. Soc. Ser. B 8 (2021), 548–577. MR 4287508, DOI 10.1090/btran/82
- Chi Li, Xiaowei Wang, and Chenyang Xu, Algebraicity of the metric tangent cones and equivariant K-stability, J. Amer. Math. Soc. 34 (2021), no. 4, 1175–1214. MR 4301561, DOI 10.1090/jams/974
- Yuchen Liu, Chenyang Xu, and Ziquan Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, Ann. of Math. (2) 196 (2022), no. 2, 507–566. MR 4445441, DOI 10.4007/annals.2022.196.2.2
- Yuji Odaka, Cristiano Spotti, and Song Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Differential Geom. 102 (2016), no. 1, 127–172. MR 3447088
- T. S. Papazachariou, K-moduli of log fano complete intersections, Preprint, arXiv:2212.09332v1, 2022.
- T. S. Papazachariou, K-moduli of log Fano_complete_intersections, PhD diss., The University of Essex (United Kingdom), 2023.
- V. V. Przhiyalkovskiĭ, I. A. Chel′tsov, and K. A. Shramov, Fano threefolds with infinite automorphism groups, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 4, 226–280 (Russian, with Russian summary); English transl., Izv. Math. 83 (2019), no. 4, 860–907. MR 3985696, DOI 10.4213/im8834
- Cristiano Spotti, Song Sun, and Chengjian Yao, Existence and deformations of Kähler-Einstein metrics on smoothable $\Bbb {Q}$-Fano varieties, Duke Math. J. 165 (2016), no. 16, 3043–3083. MR 3566198, DOI 10.1215/00127094-3645330
- Chenyang Xu, K-stability of Fano varieties: an algebro-geometric approach, EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265–354. MR 4307210, DOI 10.4171/emss/51
- Chenyang Xu and Ziquan Zhuang, On positivity of the CM line bundle on K-moduli spaces, Ann. of Math. (2) 192 (2020), no. 3, 1005–1068. MR 4172625, DOI 10.4007/annals.2020.192.3.7
- Chenyang Xu and Ziquan Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021), no. 1, 149–176. MR 4325260, DOI 10.4310/CJM.2021.v9.n1.a2
- Ziquan Zhuang, Optimal destabilizing centers and equivariant K-stability, Invent. Math. 226 (2021), no. 1, 195–223. MR 4309493, DOI 10.1007/s00222-021-01046-0
References
- Hamid Abban and Ziquan Zhuang, K-stability of Fano varieties via admissible flags, Forum Math. Pi 10 (2022), Paper No. e15, 43. MR 4448177, DOI 10.1017/fmp.2022.11
- Jarod Alper, Harold Blum, Daniel Halpern-Leistner, and Chenyang Xu, Reductivity of the automorphism group of $K$-polystable Fano varieties, Invent. Math. 222 (2020), no. 3, 995–1032. MR 4169054, DOI 10.1007/s00222-020-00987-2
- Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süß, and Nivedita Viswanathan, The Calabi problem for Fano threefolds, London Mathematical Society Lecture Note Series, vol. 485, Cambridge University Press, Cambridge, 2023. MR 4590444
- Harold Blum, Daniel Halpern-Leistner, Yuchen Liu, and Chenyang Xu, On properness of K-moduli spaces and optimal degenerations of Fano varieties, Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 73, 39. MR 4292783, DOI 10.1007/s00029-021-00694-7
- Harold Blum, Yuchen Liu, and Chenyang Xu, Openness of K-semistability for Fano varieties, Duke Math. J. 171 (2022), no. 13, 2753–2797. MR 4505846, DOI 10.1215/00127094-2022-0054
- Harold Blum and Chenyang Xu, Uniqueness of K-polystable degenerations of Fano varieties, Ann. of Math. (2) 190 (2019), no. 2, 609–656. MR 3997130, DOI 10.4007/annals.2019.190.2.4
- David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
- Ivan Cheltsov, Igor Krylov, Jesus Martinez-Garcia, and Evgeny Shinder, On maximally non-factorial nodal Fano threefolds, Rev. Mat. Iberoam. 40 (2024), no. 5, 1781–1798. MR 4792548, DOI 10.4171/rmi/1478
- Ivan Cheltsov and Jihun Park, K-stable Fano threefolds of rank 2 and degree 30, Eur. J. Math. 8 (2022), no. 3, 834–852. MR 4498816, DOI 10.1007/s40879-022-00569-x
- Ivan Cheltsov and Hendrik Süß, K-polystability of two smooth Fano threefolds, Birational geometry, Kähler-Einstein metrics and degenerations, Springer Proc. Math. Stat., vol. 409, Springer, Cham, [2023] ©2023, pp. 165–193. MR 4606635, DOI 10.1007/978-3-031-17859-7_8
- Giulio Codogni and Zsolt Patakfalvi, Positivity of the CM line bundle for families of $K$-stable klt Fano varieties, Invent. Math. 223 (2021), no. 3, 811–894. MR 4213768, DOI 10.1007/s00222-020-00999-y
- D. F. Coray and M. A. Tsfasman, Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc. (3) 57 (1988), no. 1, 25–87. MR 940430, DOI 10.1112/plms/s3-57.1.25
- Elena Denisova, On $K$-stability of $\mathbb {P}^3$ blown up along the disjoint union of a twisted cubic curve and a line, J. Lond. Math. Soc. (2) 109 (2024), no. 5, Paper No. e12911, 26. MR 4745884, DOI 10.1112/jlms.12911
- Kento Fujita, On $K$-stability and the volume functions of $\mathbb {Q}$-Fano varieties, Proc. Lond. Math. Soc. (3) 113 (2016), no. 5, 541–582. MR 3570238, DOI 10.1112/plms/pdw037
- Kento Fujita, K-stability of Fano manifolds with not small alpha invariants, J. Inst. Math. Jussieu 18 (2019), no. 3, 519–530. MR 3936640, DOI 10.1017/s1474748017000111
- Kento Fujita, Uniform K-stability and plt blowups of log Fano pairs, Kyoto J. Math. 59 (2019), no. 2, 399–418. MR 3960299, DOI 10.1215/21562261-2019-0012
- Kento Fujita, A valuative criterion for uniform K-stability of $\mathbb {Q}$-Fano varieties, J. Reine Angew. Math. 751 (2019), 309–338. MR 3956698, DOI 10.1515/crelle-2016-0055
- Kento Fujita, On K-stability for Fano threefolds of rank 3 and degree 28, Int. Math. Res. Not. IMRN 15 (2023), 12601–12784. MR 4621852, DOI 10.1093/imrn/rnac190
- Priska Jahnke and Ivo Radloff, Terminal Fano threefolds and their smoothings, Math. Z. 269 (2011), no. 3-4, 1129–1136. MR 2860279, DOI 10.1007/s00209-010-0780-8
- Anne-Sophie Kaloghiros and Andrea Petracci, On toric geometry and K-stability of Fano varieties, Trans. Amer. Math. Soc. Ser. B 8 (2021), 548–577. MR 4287508, DOI 10.1090/btran/82
- Chi Li, Xiaowei Wang, and Chenyang Xu, Algebraicity of the metric tangent cones and equivariant K-stability, J. Amer. Math. Soc. 34 (2021), no. 4, 1175–1214. MR 4301561, DOI 10.1090/jams/974
- Yuchen Liu, Chenyang Xu, and Ziquan Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, Ann. of Math. (2) 196 (2022), no. 2, 507–566. MR 4445441, DOI 10.4007/annals.2022.196.2.2
- Yuji Odaka, Cristiano Spotti, and Song Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Differential Geom. 102 (2016), no. 1, 127–172. MR 3447088
- T. S. Papazachariou, K-moduli of log fano complete intersections, Preprint, arXiv:2212.09332v1, 2022.
- T. S. Papazachariou, K-moduli of log Fano_complete_intersections, PhD diss., The University of Essex (United Kingdom), 2023.
- V. V. Przhiyalkovskiĭ, I. A. Chel′tsov, and K. A. Shramov, Fano threefolds with infinite automorphism groups, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 4, 226–280 (Russian, with Russian summary); English transl., Izv. Math. 83 (2019), no. 4, 860–907. MR 3985696, DOI 10.4213/im8834
- Cristiano Spotti, Song Sun, and Chengjian Yao, Existence and deformations of Kähler-Einstein metrics on smoothable $\mathbb {Q}$-Fano varieties, Duke Math. J. 165 (2016), no. 16, 3043–3083. MR 3566198, DOI 10.1215/00127094-3645330
- Chenyang Xu, K-stability of Fano varieties: an algebro-geometric approach, EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265–354. MR 4307210, DOI 10.4171/emss/51
- Chenyang Xu and Ziquan Zhuang, On positivity of the CM line bundle on K-moduli spaces, Ann. of Math. (2) 192 (2020), no. 3, 1005–1068. MR 4172625, DOI 10.4007/annals.2020.192.3.7
- Chenyang Xu and Ziquan Zhuang, Uniqueness of the minimizer of the normalized volume function, Camb. J. Math. 9 (2021), no. 1, 149–176. MR 4325260, DOI 10.4310/CJM.2021.v9.n1.a2
- Ziquan Zhuang, Optimal destabilizing centers and equivariant K-stability, Invent. Math. 226 (2021), no. 1, 195–223. MR 4309493, DOI 10.1007/s00222-021-01046-0
Additional Information
Hamid Abban
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
MR Author ID:
988230
ORCID:
0000-0002-4260-3243
Email:
hamid.abban@nottingham.ac.uk
Ivan Cheltsov
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
MR Author ID:
607648
Email:
i.cheltsov@ed.ac.uk
Elena Denisova
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
MR Author ID:
1604455
Email:
e.denisova@sms.ed.ac.uk
Erroxe Etxabarri-Alberdi
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Address at time of publication:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
MR Author ID:
1610439
ORCID:
0009-0001-6112-2808
Email:
Erroxe.Etxabarri-Alberdi@warwick.ac.uk
Anne-Sophie Kaloghiros
Affiliation:
Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom
MR Author ID:
912655
ORCID:
0000-0002-8305-8229
Email:
anne-sophie.kaloghiros@brunel.ac.uk
Dongchen Jiao
Affiliation:
Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom
Email:
dongchen.jiao@brunel.ac.uk
Jesus Martinez-Garcia
Affiliation:
Department of Mathematical Sciences, University of Essex, Colchester CO4 3SQ, United Kingdom
MR Author ID:
1073616
ORCID:
0000-0002-0022-371X
Email:
jesus.martinez-garcia@essex.ac.uk
Theodoros Papazachariou
Affiliation:
School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, United Kingdom
Address at time of publication:
Yau Mathematical Sciences Center, Jingzhai, Tsinghua University, Haidian District, Beijing 100084, China
ORCID:
0000-0002-2597-0201
Email:
tpapazachariou@mail.tsinghua.edu.cn
Received by editor(s):
October 13, 2023
Published electronically:
October 9, 2024
Additional Notes:
Several authors were supported by EPSRC grants: The first author was supported by EP/V048619, the second author was supported by EP/V054597/1, the fifth author was supported by EP/V056689/1, and the seventh author was supported by EP/V055399/1. The authors were supported by the University of Essex Department of Mathematical Sciences Research and Impact Fund, and the University of Essex Faculty of Science and Health Research, Innovation and Support Fund, which funded several visits to advance the project.
Article copyright:
© Copyright 2024
University Press, Inc.
