K-stability of log Fano hyperplane arrangements
Author:
Kento Fujita
Journal:
J. Algebraic Geom. 30 (2021), 603-630
DOI:
https://doi.org/10.1090/jag/783
Published electronically:
May 27, 2021
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable, or not.
References
- Valery Alexeev, Moduli of weighted hyperplane arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015. Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrìand Paolo Stellari. MR 3380944, DOI 10.1007/978-3-0348-0915-3
- Florin Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385–403. MR 2134273, DOI 10.1112/S0010437X04001071
- R. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau-Tian-Donaldson conjecture, arXiv:1509.04561v1 (2015).
- Robert J. Berman, K-polystability of ${\Bbb Q}$-Fano varieties admitting Kähler-Einstein metrics, Invent. Math. 203 (2016), no. 3, 973–1025. MR 3461370, DOI 10.1007/s00222-015-0607-7
- Sébastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841 (English, with English and French summaries). MR 3669511, DOI 10.5802/aif.3096
- Caucher Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. (2) 193 (2021), no. 2, 347–405. MR 4224714, DOI 10.4007/annals.2021.193.2.1
- Harold Blum and Mattias Jonsson, Thresholds, valuations, and K-stability, Adv. Math. 365 (2020), 107062, 57. MR 4067358, DOI 10.1016/j.aim.2020.107062
- Harold Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), no. 4, 820–849. MR 3778195, DOI 10.1112/S0010437X17008016
- Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015), no. 1, 183–197. MR 3264766, DOI 10.1090/S0894-0347-2014-00799-2
- 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
- Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2\pi$ and completion of the main proof, J. Amer. Math. Soc. 28 (2015), no. 1, 235–278. MR 3264768, DOI 10.1090/S0894-0347-2014-00801-8
- Wen Xiong Chen and Congming Li, Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991), no. 4, 359–372. MR 1129348, DOI 10.1007/BF02921311
- Sun-Yung A. Chang and Paul C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom. 27 (1988), no. 2, 259–296. MR 925123
- J.-P. Demailly, Appendix to I. Cheltsov and C. Shramov’s article, “Log canonical thresholds of smooth Fano threefolds”, On Tian’s invariant and log canonical thresholds, Russian Math. Surveys 63 (2008), no. 5, 945–950.
- Ruadhaí Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN 15 (2016), 4728–4783. MR 3564626, DOI 10.1093/imrn/rnv291
- Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511, DOI 10.1017/CBO9780511615436
- S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506, DOI 10.4310/jdg/1090950195
- 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, 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 and Yuji Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521. MR 3896135, DOI 10.2748/tmj/1546570823
- Yi Hu, Stable configurations of linear subspaces and quotient coherent sheaves, Q. J. Pure Appl. Math. 1 (2005), no. 1, 127–164. MR 2154335, DOI 10.4310/PAMQ.2005.v1.n1.a6
- Mattias Jonsson and Mircea Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209 (2013) (English, with English and French summaries). MR 3060755, DOI 10.5802/aif.2746
- George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. MR 506989, DOI 10.2307/1971168
- 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
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Chi Li, Remarks on logarithmic K-stability, Commun. Contemp. Math. 17 (2015), no. 2, 1450020, 17. MR 3313212, DOI 10.1142/S0219199714500205
- Chi Li, K-semistability is equivariant volume minimization, Duke Math. J. 166 (2017), no. 16, 3147–3218. MR 3715806, DOI 10.1215/00127094-2017-0026
- Feng Luo and Gang Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119–1129. MR 1137227, DOI 10.1090/S0002-9939-1992-1137227-5
- Chi Li and Chenyang Xu, Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180 (2014), no. 1, 197–232. MR 3194814, DOI 10.4007/annals.2014.180.1.4
- Y. Liu and Z. Zhuang, On the sharpness of Tian’s criterion for K-stability, Nagoya Math. J., DOI 10.1017/nmj.2020.28 (published online).
- Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, DOI 10.1090/S0002-9939-1988-0938672-X
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Mircea Mustaţă, Multiplier ideals of hyperplane arrangements, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5015–5023. MR 2231883, DOI 10.1090/S0002-9947-06-03895-5
- Yuji Odaka, A generalization of the Ross-Thomas slope theory, Osaka J. Math. 50 (2013), no. 1, 171–185. MR 3080636
- Yuji Odaka and Yuji Sano, Alpha invariant and K-stability of $\Bbb Q$-Fano varieties, Adv. Math. 229 (2012), no. 5, 2818–2834. MR 2889147, DOI 10.1016/j.aim.2012.01.017
- Julius Ross and Richard Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255. MR 2274514, DOI 10.1090/S1056-3911-06-00461-9
- Jacopo Stoppa, K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), no. 4, 1397–1408. MR 2518643, DOI 10.1016/j.aim.2009.02.013
- Gábor Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015), no. 1-2, 451–484. With an appendix by Sebastien Boucksom. MR 3343885, DOI 10.1007/s00208-014-1126-3
- Zach Teitler, A note on Mustaţă’s computation of multiplier ideals of hyperplane arrangements, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1575–1579. MR 2373586, DOI 10.1090/S0002-9939-07-09177-0
- 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
- Gang Tian, K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1085–1156. MR 3352459, DOI 10.1002/cpa.21578
- Marc Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. MR 1005085, DOI 10.1090/S0002-9947-1991-1005085-9
- Xiaowei Wang, Height and GIT weight, Math. Res. Lett. 19 (2012), no. 4, 909–926. MR 3008424, DOI 10.4310/MRL.2012.v19.n4.a14
References
- Valery Alexeev, Moduli of weighted hyperplane arrangements, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015. Edited by Gilberto Bini, Martí Lahoz, Emanuele Macrìand Paolo Stellari. MR 3380944, DOI 10.1007/978-3-0348-0915-3
- Florin Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385–403. MR 2134273, DOI 10.1112/S0010437X04001071
- R. Berman, S. Boucksom and M. Jonsson, A variational approach to the Yau-Tian-Donaldson conjecture, arXiv:1509.04561v1 (2015).
- Robert J. Berman, K-polystability of ${\mathbb {Q}}$-Fano varieties admitting Kähler-Einstein metrics, Invent. Math. 203 (2016), no. 3, 973–1025. MR 3461370, DOI 10.1007/s00222-015-0607-7
- Sébastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841 (English, with English and French summaries). MR 3669511
- Caucher Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann. of Math. (2) 193 (2021), no. 2, 347–405. MR 4224714, DOI 10.4007/annals.2021.193.2.1
- Harold Blum and Mattias Jonsson, Thresholds, valuations, and K-stability, Adv. Math. 365 (2020), 107062, 57. MR 4067358, DOI 10.1016/j.aim.2020.107062
- Harold Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), no. 4, 820–849. MR 3778195, DOI 10.1112/S0010437X17008016
- Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015), no. 1, 183–197. MR 3264766, DOI 10.1090/S0894-0347-2014-00799-2
- 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
- Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2\pi$ and completion of the main proof, J. Amer. Math. Soc. 28 (2015), no. 1, 235–278. MR 3264768, DOI 10.1090/S0894-0347-2014-00801-8
- Wen Xiong Chen and Congming Li, Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991), no. 4, 359–372. MR 1129348, DOI 10.1007/BF02921311
- Sun-Yung A. Chang and Paul C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom. 27 (1988), no. 2, 259–296. MR 925123
- J.-P. Demailly, Appendix to I. Cheltsov and C. Shramov’s article, “Log canonical thresholds of smooth Fano threefolds”, On Tian’s invariant and log canonical thresholds, Russian Math. Surveys 63 (2008), no. 5, 945–950.
- Ruadhaí Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN 15 (2016), 4728–4783. MR 3564626, DOI 10.1093/imrn/rnv291
- Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511, DOI 10.1017/CBO9780511615436
- S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506
- 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, 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 and Yuji Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521. MR 3896135, DOI 10.2748/tmj/1546570823
- Yi Hu, Stable configurations of linear subspaces and quotient coherent sheaves, Q. J. Pure Appl. Math. 1 (2005), no. 1, 127–164. MR 2154335
- Mattias Jonsson and Mircea Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209 (2013) (English, with English and French summaries). MR 3060755, DOI 10.5802/aif.2746
- George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. MR 506989, DOI 10.2307/1971168
- 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
- Robert Lazarsfeld, Positivity in algebraic geometry. I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. MR 2095471, DOI 10.1007/978-3-642-18808-4
- Robert Lazarsfeld, Positivity in algebraic geometry. II: Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Chi Li, Remarks on logarithmic K-stability, Commun. Contemp. Math. 17 (2015), no. 2, 1450020, 17. MR 3313212, DOI 10.1142/S0219199714500205
- Chi Li, K-semistability is equivariant volume minimization, Duke Math. J. 166 (2017), no. 16, 3147–3218. MR 3715806, DOI 10.1215/00127094-2017-0026
- Feng Luo and Gang Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119–1129. MR 1137227, DOI 10.2307/2159498
- Chi Li and Chenyang Xu, Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180 (2014), no. 1, 197–232. MR 3194814, DOI 10.4007/annals.2014.180.1.4
- Y. Liu and Z. Zhuang, On the sharpness of Tian’s criterion for K-stability, Nagoya Math. J., DOI 10.1017/nmj.2020.28 (published online).
- Robert C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222–224. MR 938672, DOI 10.2307/2047555
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Mircea Mustaţă, Multiplier ideals of hyperplane arrangements, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5015–5023. MR 2231883, DOI 10.1090/S0002-9947-06-03895-5
- Yuji Odaka, A generalization of the Ross-Thomas slope theory, Osaka J. Math. 50 (2013), no. 1, 171–185. MR 3080636
- Yuji Odaka and Yuji Sano, Alpha invariant and K-stability of $\mathbb {Q}$-Fano varieties, Adv. Math. 229 (2012), no. 5, 2818–2834. MR 2889147, DOI 10.1016/j.aim.2012.01.017
- Julius Ross and Richard Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255. MR 2274514, DOI 10.1090/S1056-3911-06-00461-9
- Jacopo Stoppa, K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), no. 4, 1397–1408. MR 2518643, DOI 10.1016/j.aim.2009.02.013
- Gábor Székelyhidi, Filtrations and test-configurations, Math. Ann. 362 (2015), no. 1-2, 451–484. With an appendix by Sebastien Boucksom. MR 3343885, DOI 10.1007/s00208-014-1126-3
- Zach Teitler, A note on Mustaţă’s computation of multiplier ideals of hyperplane arrangements, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1575–1579. MR 2373586, DOI 10.1090/S0002-9939-07-09177-0
- 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
- Gang Tian, K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1085–1156. MR 3352459, DOI 10.1002/cpa.21578
- Marc Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. MR 1005085, DOI 10.2307/2001742
- Xiaowei Wang, Height and GIT weight, Math. Res. Lett. 19 (2012), no. 4, 909–926. MR 3008424, DOI 10.4310/MRL.2012.v19.n4.a14
Additional Information
Kento Fujita
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
MR Author ID:
962140
ORCID:
0000-0002-8887-5551
Email:
fujita@math.sci.osaka-u.ac.jp
Received by editor(s):
September 24, 2017
Received by editor(s) in revised form:
July 1, 2019, and February 10, 2021
Published electronically:
May 27, 2021
Additional Notes:
This work was supported by JSPS KAKENHI Grant Number JP16H06885
Article copyright:
© Copyright 2021
University Press, Inc.