We study logarithmic K-stability for pairs by extending the formula for Donaldson-Futaki invariants to log setting. We also provide algebro-geometric counterparts of recent results of existence of Kähler-Einstein metrics with cone singularities. In particular, we will study the relation between log K-stability and the global log canonical thresholds.
Nous étudions la K-stabilité logarithmique pour des paires, étendant la formule pour les invariants de Donaldson-Futaki au contexte logarithmique. Nous développons également le versant algébro-géométrique de résultats récents d’existence de métriques Kähler-Einstein à singularités coniques. Nous étudions notamment la relation entre la stabilité logarithmique et les seuils log canoniques globaux.
@article{AFST_2015_6_24_3_505_0,
author = {Yuji Odaka and Song Sun},
title = {Testing {Log} {K-stability} by blowing up formalism},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {505--522},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 24},
number = {3},
year = {2015},
doi = {10.5802/afst.1453},
mrnumber = {3403730},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1453/}
}
TY - JOUR AU - Yuji Odaka AU - Song Sun TI - Testing Log K-stability by blowing up formalism JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2015 SP - 505 EP - 522 VL - 24 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1453/ DO - 10.5802/afst.1453 LA - en ID - AFST_2015_6_24_3_505_0 ER -
%0 Journal Article %A Yuji Odaka %A Song Sun %T Testing Log K-stability by blowing up formalism %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2015 %P 505-522 %V 24 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1453/ %R 10.5802/afst.1453 %G en %F AFST_2015_6_24_3_505_0
Yuji Odaka; Song Sun. Testing Log K-stability by blowing up formalism. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 3, pp. 505-522. doi : 10.5802/afst.1453. https://afst.centre-mersenne.org/articles/10.5802/afst.1453/
[1] Ambro (F.).— Quasi-log varieties, Tr. Mat. Inst. Steklova vol. 240 p. 220-239 (2003); English translation in Proc. Steklov Inst. Math. vol. 240, p. 214-233 (2003). | MR | Zbl
[2] Ambro (F.).— The moduli -divisor of an lc-trivial fibration. (English summary) Compos. Math. vol. 141, p. 385-403 (2005). | MR | Zbl
[3] Berman (R. J.).— A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics. Adv. Math. 248, p. 1254-1297 (2013). | MR | Zbl
[4] Berman (R.).— K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics, to appear in Invent. Math.
[5] Brendle (S.).— Ricci-flat Kähler metrics with edge singularities, Int. Math. Res. Not. IMRN, no. 24, p. 5727-5766 (2013). | MR | Zbl
[6] Chen (X-X.), Donaldson (S.), Sun (S.).— Kähler-Einstein metrics on Fano manifolds, I, II, III. J. Amer. Math. Soc. 28, p. 183-278 (2015). | MR
[7] Cheltsov (I. A.), Shramov (K. A.).— Log canonical thresholds of smooth Fano threefolds, With an appendix by Jean-Pierre Demailly, English translation: Russian Mathematical Surveys, vol. 63, no. 5, p. 859-958 (2008). | MR | Zbl
[8] Donaldson (S.).— Scalar curvature and stability of toric varieties, J. Diff. Geom. (2002). | MR | Zbl
[9] Donaldson (S. K.).— Kähler metrics with cone singularities along a divisor. Essays in mathematics and its applications, 49-79, Springer, Heidelberg (2012). | MR
[10] Fujino (O.).— Non-vanishing theorems for log canonical pairs, J. Alg. Geom. vol. 20, p. 771-783 (2011). | MR | Zbl
[11] Jeffres (T.), Mazzeo (R.), Rubinstein (Y.).— Kähler-Einstein metrics with edge singularities, arxiv: 1105.5216.
[12] Kawakita (M.).— Inversion of adjunction on log canonicity, Invent. Math. vol. 167, p. 129-133, (2007). | MR | Zbl
[13] Kollár (J.), Mori (S.).— Birational geometry of algebraic varieties, Cambridge University Press (1998). | MR | Zbl
[14] Li (C.).— Remarks on logarithmic K-stability, Communications in Contemporary Mathematics Vol. 17, No. 2 (2015). | MR
[15] Li (C.), Xu (C.).— Special test configurations and K-stability of Fano varieties, Ann. of Math. (2) 180, no. 1, p. 197-232 (2014). | MR | Zbl
[16] Mumford (D.).— Stability of Projective Varieties, Enseignement Math. vol. 23 (1977). | MR | Zbl
[17] Odaka (Y.).— On the GIT stability of polarized varieties via Discrepancy, Ann. of Math. (2) 177, no. 2, p. 645-661 (2013). | MR | Zbl
[18] Odaka (Y.).— A generalization of Ross-Thomas slope theory, Osaka J. Math. 50, no. 1, p. 171-185 (2013). | MR
[19] Odaka (Y.).— The Calabi conjecture and K-stability, Int. Math. Res. Not. IMRN, no. 10, p. 2272-2288 (2012). | MR
[20] Odaka (Y.).— On the K-stability of (-)Fano varieties, slide of the talk at Kyoto Symposium, Complex analysis, July (2011). available at http://www.kurims.kyoto-u.ac.jp/ yodaka/Kyoto.Hayama.pdf
[21] Odaka (Y.), Y. Sano.— Alpha invariants and K-stability of -Fano varieties, Adv. Math. 229, no. 5, p. 2818-2834 (2012). | MR | Zbl
[22] Odaka (Y.), Xu (C.).— Log canonical models of singular pairs and its applications, Math. Res. Notices. vol. 19 (2012) | MR | Zbl
[23] Ross (J.), Thomas (R.).— A study of Hilbert-Mumford criterion of stability of projective varieties, J. Alg. Geom. (2007). | MR
[24] Stoppa (J.).— A note on the definition of K-stability, arXiv:1111.5826 (2011).
[25] Sun (S.).— Note on K-stability of pairs, Math. Ann. 355, no. 1, p. 259-272 (2013). | MR | Zbl
[26] Tian (G.).— On Kähler-Einstein metrics on certain Kähler manifolds with , Invent. Math. vol. 89, p. 225-246 (1987). | MR | Zbl
[27] Tian (G.).— Kähler-Einstein metrics with positive scalar curvature, Invent. Math. vol. 130, p. 1-37, (1997). | MR | Zbl
[28] Wang (X.).— Heights and GIT weights, Math. Res. Notices vol. 19 (2012).
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