Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T09:02:04.408Z Has data issue: false hasContentIssue false
  • English
  • Français

K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS

Part of: Surfaces and higher-dimensional varieties Algebraic groups

Published online by Cambridge University Press:  30 March 2017

Kento Fujita
Kento Fujita*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that any $n$-dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

Keywords

Fano varietiesK-stabilityKähler–Einstein metrics

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14L24: Geometric invariant theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 3 , May 2019 , pp. 519 - 530
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arezzo, C., Ghigi, A. and Pirola, G. P., Symmetries, quotients and Kähler–Einstein metrics, J. Reine Angew. Math. 591 (2006), 177–200.Google Scholar
Berman, R., K-polystability of Q-Fano varieties admitting Kähler–Einstein metrics, Invent. Math. 203(3) (2016), 973–1025.Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18(2) (2009), 279–308.Google Scholar
Boucksom, S., Hisamoto, T. and Jonsson, M., Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, preprint, 2015,arXiv:1504.06568.Google Scholar
Chen, X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities, J. Amer. Math. Soc. 28(1) (2015), 183–197.Google Scholar
Chen, X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds, II: limits with cone angle less than 2𝜋, J. Amer. Math. Soc. 28(1) (2015), 199–234.Google Scholar
Chen, X., Donaldson, S. and Sun, S., Kähler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2𝜋 and completion of the main proof, J. Amer. Math. Soc. 28(1) (2015), 235–278.Google Scholar
Chel’tsov, I. A., Log canonical thresholds on hypersurfaces, Sb. Math. 192(7–8) (2001), 1241–1257.Google Scholar
Chel’tsov, I. A., Log canonical thresholds of del Pezzo surfaces, Geom. Funct. Anal. 18(4) (2008), 1118–1144.Google Scholar
Chel’tsov, I. A. and Park, J., Global log-canonical thresholds and generalized Eckardt points, Sb. Math. 193(5–6) (2002), 779–789.Google Scholar
Chel’tsov, I. A., Park, J. and Won, J., Log canonical thresholds of certain Fano hypersurfaces, Math. Z. 276(1–2) (2014), 51–79.Google Scholar
Chel’tsov, I. A. and Shramov, K. A., Log-canonical thresholds for nonsingular Fano threefolds, Russian Math. Surveys 63(5) (2008), 859–958.Google Scholar
Demailly, J.-P., 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(5) (2008), 945–950.Google Scholar
Dervan, R., Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN 2016(15) 4728–4783.Google Scholar
Dervan, R., On K-stability of finite covers, Bull. Lond. Math. Soc. 48(4) (2016), 717–728.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett. 10(2–3) (2003), 219–236.Google Scholar
Ding, W. and Tian, G., Kähler–Einstein metrics and the generalized Futaki invariants, Invent. Math. 110(2) (1992), 315–335.Google Scholar
Donaldson, S., Scalar curvature and stability of toric varieties, J. Differential Geom. 62(2) (2002), 289–349.Google Scholar
Donaldson, S., Lower bounds on the Calabi functional, J. Differential Geom. 70(3) (2005), 453–472.Google Scholar
Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M. and Popa, M., Restricted volumes and base loci of linear series, Amer. J. Math. 131(3) (2009), 607–651.Google Scholar
Fujita, K. and Odaka, Y., On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (accepted) arXiv:1602.01305v2.Google Scholar
Fujita, K., On K-stability and the volume functions of ℚ-Fano varieties, Proc. Lond. Math. Soc. 113(5) (2016), 541–582.Google Scholar
Fujita, K., A valuative criterion for uniform K-stability of ℚ-Fano varieties, J. Reine Angew. Math. doi:10.1515/crelle-2016-0055.Google Scholar
Fulton, W., Introduction to toric varieties, in Annals of Mathematics Studies, The William H. Roever Lectures in Geometry, Volume 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Kaloghiros, A.-S., Küronya, A. and Lazić, V., Finite generation and geography of models, in Minimal Models and Extremal Rays, Kyoto, 2011, Advanced Studies in Pure Mathematics, Volume 70, pp. 215–254 (Mathematical Society of Japan, Tokyo, 2016).Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, in With the collaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Math., 134 (Cambridge University Press, Cambridge, 1998).Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, in Ergebnisse der Mathematik und ihrer Grenzgebiete. (3), 48 (Springer, Berlin, 2004).Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry, II: Positivity for Vector Bundles, and Multiplier Ideals, in Ergebnisse der Mathematik und ihrer Grenzgebiete. (3), 49 (Springer, Berlin, 2004).Google Scholar
Li, C., Minimizing normalized volumes of valuations, preprint, 2015, arXiv:1511.08164.Google Scholar
Li, C., K-semistability is equivariant volume minimization, preprint, 2015,arXiv:1512.07205v3.Google Scholar
Liu, Y., The volume of singular Kähler–Einstein Fano varieties, preprint, 2016,arXiv:1605.01034.Google Scholar
Lazarsfeld, R. and Mustaţă, M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42(5) (2009), 783–835.Google Scholar
Li, C. and Xu, C., Special test configuration and K-stability of Fano varieties, Ann. of Math. (2) 180(1) (2014), 197–232.Google Scholar
Lu, Z., On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math. 122(2) (2000), 235–273.Google Scholar
Mabuchi, T., K-stability of constant scalar curvature, preprint, 2008, arXiv:0812.4903.Google Scholar
Mabuchi, T., A stronger concept of K-stability, preprint, 2009, arXiv:0910.4617.Google Scholar
Matsushima, Y., Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J. 11 (1957), 145–150.Google Scholar
Nadel, A. M., Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132(3) (1990), 549–596.Google Scholar
Odaka, Y., A generalization of the Ross–Thomas slope theory, Osaka J. Math. 50(1) (2013), 171–185.Google Scholar
Odaka, Y. and Sano, Y., Alpha invariants and K-stability of ℚ-Fano varieties, Adv. Math. 229(5) (2012), 2818–2834.Google Scholar
Park, J., Birational maps of del Pezzo fibrations, J. Reine Angew. Math. 538 (2001), 213–221.Google Scholar
Pukhlikov, A. V., Birational geometry of Fano direct products, Izv. Math. 69(6) (2005), 1225–1255.Google Scholar
Ross, J. and Thomas, R., A study of the Hilbert–Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16(2) (2007), 201–255.Google Scholar
Stoppa, J., K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221(4) (2009), 1397–1408.Google Scholar
Tian, G., On Kähler–Einstein metrics on certain Kähler manifolds with C 1(M) > 0, Invent. Math. 89(2) (1987), 225–246.+0,+Invent.+Math.+89(2)+(1987),+225–246.>Google Scholar
Tian, G., On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101(1) (1990), 101–172.Google Scholar
Tian, G., Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130(1) (1997), 1–37.Google Scholar
Tian, G., K-stability and Kähler–Einstein metrics, Comm. Pure Appl. Math. 68(7) (2015), 1085–1156.Google Scholar
Tian, G. and Yau, S. T., Kähler–Einstein metrics on complex surfaces with C 1 > 0, Comm. Math. Phys. 112(1) (1987), 175–203.+0,+Comm.+Math.+Phys.+112(1)+(1987),+175–203.>Google Scholar
Wang, X., Height and GIT weight, Math. Res. Lett. 19(4) (2012), 909–926.Google Scholar
Zelditch, S., Szegö kernels and a theorem of Tian, Int. Math. Res. Not. IMRN 1998(6) 317–331.Google Scholar