Kähler–Einstein Fano threefolds of degree $22$

Authors:
Ivan Cheltsov and Constantin Shramov

Journal:
J. Algebraic Geom. **32** (2023), 385-428

DOI:
https://doi.org/10.1090/jag/812

Published electronically:
March 15, 2023

Full-text PDF

Abstract |
References |
Additional Information

Abstract: We study the problem of existence of Kähler–Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree $22$ that admit a faithful action of the multiplicative group $\mathbb {C}^\ast$. We prove that, with the possible exception of two explicitly described cases, all such smooth Fano threefolds are Kähler–Einstein.

References
- Maxim Arap, Joseph Cutrone, and Nicholas Marshburn,
*On the existence of certain weak Fano threefolds of Picard number two*, Math. Scand. **120** (2017), no. 1, 68–86. MR **3624007**, DOI 10.7146/math.scand.a-25505
- C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. Martinez-Garcia, C. Shramov, H. Süß, and N. Viswanathan,
*The Calabi problem for Fano threefolds*, MPIM preprint 2021-31.
- H. Blum, Yu. Liu, and Ch. Xu,
*Openness of K-semistability for Fano varieties*, arXiv:1907.02408, 2019.
- Ivan Cheltsov, Jihun Park, and Joonyeong Won,
*Affine cones over smooth cubic surfaces*, J. Eur. Math. Soc. (JEMS) **18** (2016), no. 7, 1537–1564. MR **3506607**, DOI 10.4171/JEMS/622
- I. A. Chel′tsov and K. A. Shramov,
*Log-canonical thresholds for nonsingular Fano threefolds*, Uspekhi Mat. Nauk **63** (2008), no. 5(383), 73–180 (Russian, with Russian summary); English transl., Russian Math. Surveys **63** (2008), no. 5, 859–958. MR **2484031**, DOI 10.1070/RM2008v063n05ABEH004561
- I. A. Chel′tsov and K. A. Shramov,
*Extremal metrics on del Pezzo threefolds*, Tr. Mat. Inst. Steklova **264** (2009), no. Mnogomernaya Algebraicheskaya Geometriya, 37–51 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. **264** (2009), no. 1, 30–44. MR **2590832**, DOI 10.1134/S0081543809010040
- Ivan Cheltsov and Constantin Shramov,
*Three embeddings of the Klein simple group into the Cremona group of rank three*, Transform. Groups **17** (2012), no. 2, 303–350. MR **2921069**, DOI 10.1007/s00031-012-9183-8
- Ivan Cheltsov and Constantin Shramov,
*Five embeddings of one simple group*, Trans. Amer. Math. Soc. **366** (2014), no. 3, 1289–1331. MR **3145732**, DOI 10.1090/S0002-9947-2013-05768-6
- Ivan Cheltsov and Constantin Shramov,
*Two rational nodal quartic 3-folds*, Q. J. Math. **67** (2016), no. 4, 573–601. MR **3609847**, DOI 10.1093/qmath/haw032
- Ivan Cheltsov and Constantin Shramov,
*Cremona groups and the icosahedron*, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR **3444095**
- Ivan Cheltsov and Constantin Shramov,
*Finite collineation groups and birational rigidity*, Selecta Math. (N.S.) **25** (2019), no. 5, Paper No. 71, 68. MR **4036497**, DOI 10.1007/s00029-019-0516-5
- Xiuxiong Chen, Simon Donaldson, and Song Sun,
*Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $2\pi$*, J. Amer. Math. Soc. **28** (2015), no. 1, 199–234. MR **3264767**, DOI 10.1090/S0894-0347-2014-00800-6
- Alessio Corti,
*Factoring birational maps of threefolds after Sarkisov*, J. Algebraic Geom. **4** (1995), no. 2, 223–254. MR **1311348**
- Joseph W. Cutrone and Nicholas A. Marshburn,
*Towards the classification of weak Fano threefolds with $\rho =2$*, Cent. Eur. J. Math. **11** (2013), no. 9, 1552–1576. MR **3071923**, DOI 10.2478/s11533-013-0261-5
- Sławomir Dinew, Grzegorz Kapustka, and MichałKapustka,
*Remarks on Mukai threefolds admitting $\Bbb C^*$ action*, Mosc. Math. J. **17** (2017), no. 1, 15–33. MR **3634518**, DOI 10.17323/1609-4514-2017-17-1-15-33
- Simon K. Donaldson,
*Kähler geometry on toric manifolds, and some other manifolds with large symmetry*, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 29–75. MR **2483362**
- Simon K. Donaldson,
*Stability of algebraic varieties and Kähler geometry*, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 199–221. MR **3821150**, DOI 10.1090/pspum/097.1/01673
- K. Fujita,
*On Fano threefolds of degree $22$ after Cheltsov and Shramov*, preprint, arXiv:2107.04816, 2021.
- V. Iskovskikh,
*Fano 3-folds I*, Math. U.S.S.R. Izvestiya **11** (1977), 485–527.
- V. A. Iskovskih,
*Fano threefolds. II*, Izv. Akad. Nauk SSSR Ser. Mat. **42** (1978), no. 3, 506–549 (Russian). MR **503430**
- V. A. Iskovskikh and Yu. G. Prokhorov,
*Fano varieties*, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR **1668579**
- Yujiro Kawamata,
*On Fujita’s freeness conjecture for $3$-folds and $4$-folds*, Math. Ann. **308** (1997), no. 3, 491–505. MR **1457742**, DOI 10.1007/s002080050085
- Yujiro Kawamata,
*Subadjunction of log canonical divisors. II*, Amer. J. Math. **120** (1998), no. 5, 893–899. MR **1646046**, DOI 10.1353/ajm.1998.0038
- János Kollár,
*Singularities of pairs*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR **1492525**, DOI 10.1090/pspum/062.1/1492525
- János Kollár,
*Lectures on resolution of singularities*, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR **2289519**
- János Kollár and Shigefumi Mori,
*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR **1658959**, DOI 10.1017/CBO9780511662560
- Alexander Kuznetsov and Yuri Prokhorov,
*Prime Fano threefolds of genus 12 with a $\Bbb G_\textrm {m}$-action and their automorphisms*, Épijournal Géom. Algébrique **2** (2018), Art. 3, 14. MR **3816899**, DOI 10.46298/epiga.2018.volume2.4179
- Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov,
*Hilbert schemes of lines and conics and automorphism groups of Fano threefolds*, Jpn. J. Math. **13** (2018), no. 1, 109–185. MR **3776469**, DOI 10.1007/s11537-017-1714-6
- Shigeru Mukai and Hiroshi Umemura,
*Minimal rational threefolds*, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 490–518. MR **726439**, DOI 10.1007/BFb0099976
- Yu. G. Prokhorov,
*Automorphism groups of Fano $3$-folds*, Uspekhi Mat. Nauk **45** (1990), no. 3(273), 195–196 (Russian); English transl., Russian Math. Surveys **45** (1990), no. 3, 222–223. MR **1071944**, DOI 10.1070/RM1990v045n03ABEH002363
- Yuri Prokhorov,
*$G$-Fano threefolds, II*, Adv. Geom. **13** (2013), no. 3, 419–434. MR **3100918**, DOI 10.1515/advgeom-2013-0009
- Yu. G. Prokhorov,
*Singular Fano manifolds of genus 12*, Mat. Sb. **207** (2016), no. 7, 101–130 (Russian, with Russian summary); English transl., Sb. Math. **207** (2016), no. 7-8, 983–1009. MR **3535377**, DOI 10.4213/sm8585
- Z. Reichstein and B. Youssin,
*Equivariant resolution of points of indeterminacy*, Proc. Amer. Math. Soc. **130** (2002), no. 8, 2183–2187. MR **1896397**, DOI 10.1090/S0002-9939-02-06595-4
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR **715649**, DOI 10.2969/aspm/00110131
- Yann Rollin, Santiago R. Simanca, and Carl Tipler,
*Deformation of extremal metrics, complex manifolds and the relative Futaki invariant*, Math. Z. **273** (2013), no. 1-2, 547–568. MR **3010175**, DOI 10.1007/s00209-012-1019-7
- Kiyohiko Takeuchi,
*Some birational maps of Fano $3$-folds*, Compositio Math. **71** (1989), no. 3, 265–283. MR **1022045**
- Gang Tian,
*On Kähler-Einstein metrics on certain Kähler manifolds with $C_1(M)>0$*, Invent. Math. **89** (1987), no. 2, 225–246. MR **894378**, DOI 10.1007/BF01389077
- Gang Tian,
*Kähler-Einstein metrics with positive scalar curvature*, Invent. Math. **130** (1997), no. 1, 1–37. MR **1471884**, DOI 10.1007/s002220050176
- Chenyang Xu,
*A minimizing valuation is quasi-monomial*, Ann. of Math. (2) **191** (2020), no. 3, 1003–1030. MR **4088355**, DOI 10.4007/annals.2020.191.3.6

References
- Maxim Arap, Joseph Cutrone, and Nicholas Marshburn,
*On the existence of certain weak Fano threefolds of Picard number two*, Math. Scand. **120** (2017), no. 1, 68–86. MR **3624007**, DOI 10.7146/math.scand.a-25505
- C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. Martinez-Garcia, C. Shramov, H. Süß, and N. Viswanathan,
*The Calabi problem for Fano threefolds*, MPIM preprint 2021-31.
- H. Blum, Yu. Liu, and Ch. Xu,
*Openness of K-semistability for Fano varieties*, arXiv:1907.02408, 2019.
- Ivan Cheltsov, Jihun Park, and Joonyeong Won,
*Affine cones over smooth cubic surfaces*, J. Eur. Math. Soc. (JEMS) **18** (2016), no. 7, 1537–1564. MR **3506607**, DOI 10.4171/JEMS/622
- I. A. Cheltsov and K. A. Shramov,
*Log-canonical thresholds for nonsingular Fano threefolds*, Uspekhi Mat. Nauk **63** (2008), no. 5(383), 73–180 (Russian, with Russian summary); English transl., Russian Math. Surveys **63** (2008), no. 5, 859–958. MR **2484031**, DOI 10.1070/RM2008v063n05ABEH004561
- I. A. Cheltsov and K. A. Shramov,
*Extremal metrics on del Pezzo threefolds*, Tr. Mat. Inst. Steklova **264** (2009), no. Mnogomernaya Algebraicheskaya Geometriya, 37–51 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. **264** (2009), no. 1, 30–44. MR **2590832**, DOI 10.1134/S0081543809010040
- Ivan Cheltsov and Constantin Shramov,
*Three embeddings of the Klein simple group into the Cremona group of rank three*, Transform. Groups **17** (2012), no. 2, 303–350. MR **2921069**, DOI 10.1007/s00031-012-9183-8
- Ivan Cheltsov and Constantin Shramov,
*Five embeddings of one simple group*, Trans. Amer. Math. Soc. **366** (2014), no. 3, 1289–1331. MR **3145732**, DOI 10.1090/S0002-9947-2013-05768-6
- Ivan Cheltsov and Constantin Shramov,
*Two rational nodal quartic 3-folds*, Q. J. Math. **67** (2016), no. 4, 573–601. MR **3609847**, DOI 10.1093/qmath/haw032
- Ivan Cheltsov and Constantin Shramov,
*Cremona groups and the icosahedron*, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR **3444095**
- Ivan Cheltsov and Constantin Shramov,
*Finite collineation groups and birational rigidity*, Selecta Math. (N.S.) **25** (2019), no. 5, Paper No. 71, 68. MR **4036497**, DOI 10.1007/s00029-019-0516-5
- Xiuxiong Chen, Simon Donaldson, and Song Sun,
*Kähler-Einstein metrics on Fano manifolds. I, II, III*, J. Amer. Math. Soc. **28** (2015), no. 1, 183–197, 199–234, 235–278. MR **3264767**, DOI 10.1090/S0894-0347-2014-00800-6
- Alessio Corti,
*Factoring birational maps of threefolds after Sarkisov*, J. Algebraic Geom. **4** (1995), no. 2, 223–254. MR **1311348**
- Joseph W. Cutrone and Nicholas A. Marshburn,
*Towards the classification of weak Fano threefolds with $\rho =2$*, Cent. Eur. J. Math. **11** (2013), no. 9, 1552–1576. MR **3071923**, DOI 10.2478/s11533-013-0261-5
- Sławomir Dinew, Grzegorz Kapustka, and MichałKapustka,
*Remarks on Mukai threefolds admitting $\mathbb {C}^*$ action*, Mosc. Math. J. **17** (2017), no. 1, 15–33. MR **3634518**, DOI 10.17323/1609-4514-2017-17-1-15-33
- Simon K. Donaldson,
*Kähler geometry on toric manifolds, and some other manifolds with large symmetry*, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 29–75. MR **2483362**
- Simon K. Donaldson,
*Stability of algebraic varieties and Kähler geometry*, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 199–221. MR **3821150**, DOI 10.4310/pamq.2009.v5.n2.a2
- K. Fujita,
*On Fano threefolds of degree $22$ after Cheltsov and Shramov*, preprint, arXiv:2107.04816, 2021.
- V. Iskovskikh,
*Fano 3-folds I*, Math. U.S.S.R. Izvestiya **11** (1977), 485–527.
- V. A. Iskovskih,
*Fano threefolds. II*, Izv. Akad. Nauk SSSR Ser. Mat. **42** (1978), no. 3, 506–549 (Russian). MR **503430**
- V. A. Iskovskikh and Yu. G. Prokhorov,
*Fano varieties*, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR **1668579**
- Yujiro Kawamata,
*On Fujita’s freeness conjecture for $3$-folds and $4$-folds*, Math. Ann. **308** (1997), no. 3, 491–505. MR **1457742**, DOI 10.1007/s002080050085
- Yujiro Kawamata,
*Subadjunction of log canonical divisors. II*, Amer. J. Math. **120** (1998), no. 5, 893–899. MR **1646046**
- János Kollár,
*Singularities of pairs*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR **1492525**
- János Kollár,
*Lectures on resolution of singularities*, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR **2289519**
- János Kollár and Shigefumi Mori,
*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR **1658959**, DOI 10.1017/CBO9780511662560
- Alexander Kuznetsov and Yuri Prokhorov,
*Prime Fano threefolds of genus 12 with a $\mathbb {G}_{\mathrm {m}}$-action and their automorphisms*, Épijournal Géom. Algébrique **2** (2018), Art. 3, 14. MR **3816899**, DOI 10.46298/epiga.2018.volume2.4179
- Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov,
*Hilbert schemes of lines and conics and automorphism groups of Fano threefolds*, Jpn. J. Math. **13** (2018), no. 1, 109–185. MR **3776469**, DOI 10.1007/s11537-017-1714-6
- Shigeru Mukai and Hiroshi Umemura,
*Minimal rational threefolds*, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 490–518. MR **726439**, DOI 10.1007/BFb0099976
- Yu. G. Prokhorov,
*Automorphism groups of Fano $3$-folds*, Uspekhi Mat. Nauk **45** (1990), no. 3(273), 195–196 (Russian); English transl., Russian Math. Surveys **45** (1990), no. 3, 222–223. MR **1071944**, DOI 10.1070/RM1990v045n03ABEH002363
- Yuri Prokhorov,
*$G$-Fano threefolds, II*, Adv. Geom. **13** (2013), no. 3, 419–434. MR **3100918**, DOI 10.1515/advgeom-2013-0009
- Yu. G. Prokhorov,
*Singular Fano manifolds of genus 12*, Mat. Sb. **207** (2016), no. 7, 101–130 (Russian, with Russian summary); English transl., Sb. Math. **207** (2016), no. 7-8, 983–1009. MR **3535377**, DOI 10.4213/sm8585
- Z. Reichstein and B. Youssin,
*Equivariant resolution of points of indeterminacy*, Proc. Amer. Math. Soc. **130** (2002), no. 8, 2183–2187. MR **1896397**, DOI 10.1090/S0002-9939-02-06595-4
- Miles Reid,
*Minimal models of canonical $3$-folds*, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR **715649**, DOI 10.2969/aspm/00110131
- Yann Rollin, Santiago R. Simanca, and Carl Tipler,
*Deformation of extremal metrics, complex manifolds and the relative Futaki invariant*, Math. Z. **273** (2013), no. 1-2, 547–568. MR **3010175**, DOI 10.1007/s00209-012-1019-7
- Kiyohiko Takeuchi,
*Some birational maps of Fano $3$-folds*, Compositio Math. **71** (1989), no. 3, 265–283. MR **1022045**
- Gang Tian,
*On Kähler-Einstein metrics on certain Kähler manifolds with $C_1(M)>0$*, Invent. Math. **89** (1987), no. 2, 225–246. MR **894378**, DOI 10.1007/BF01389077
- Gang Tian,
*Kähler-Einstein metrics with positive scalar curvature*, Invent. Math. **130** (1997), no. 1, 1–37. MR **1471884**, DOI 10.1007/s002220050176
- Chenyang Xu,
*A minimizing valuation is quasi-monomial*, Ann. of Math. (2) **191** (2020), no. 3, 1003–1030. MR **4088355**, DOI 10.4007/annals.2020.191.3.6

Additional Information

**Ivan Cheltsov**

Affiliation:
School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

MR Author ID:
607648

ORCID:
0000-0002-6820-8073

Email:
I.Cheltsov@ed.ac.uk

**Constantin Shramov**

Affiliation:
Steklov Mathematical Institute of RAS, 8 Gubkina Street, Moscow 119991, Russia; and National Research University Higher School of Economics, Laboratory of Algebraic Geometry, NRU HSE, 6 Usacheva str., Moscow, 117312, Russia

MR Author ID:
907948

Email:
costya.shramov@gmail.com

Received by editor(s):
May 26, 2020

Published electronically:
March 15, 2023

Additional Notes:

The work of the first author has been supported by EPSRC

grant number EP/V054597/1. The work of the second author was performed at the Steklov International Mathematical Center and was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265). He was also supported by the Russian Academic Excellence Project “5-100” and the Young Russian Mathematicians award.

Article copyright:
© Copyright 2023
University Press, Inc.