Log-canonical thresholds in real and complex dimension 2

Collins, Tristan C.

Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2883-2900.

Nous étudions l’ensemble des seuils log-canoniques (ou indices d’intégrabilité critiques) des germes de fonctions holomorphes (resp. réel analytiques) dans ${ℂ}^2$ (resp. ${ℝ}^2$). En particulier, nous prouvons que la condition de la chaîne ascendante est vraie et que les points d’accumulation positifs des séquences décroissantes sont précisément les indices d’intégrabilité des fonctions holomorphes (resp. réel analytiques) en dimension $1$. Cela donne une nouvelle preuve d’un théorème de Phong-Sturm.

We study the set of log-canonical thresholds (or critical integrability indices) of holomorphic (resp. real analytic) function germs in ${ℂ}^2$ (resp. ${ℝ}^2$). In particular, we prove that the ascending chain condition holds, and that the positive accumulation points of decreasing sequences are precisely the integrability indices of holomorphic (resp. real analytic) functions in dimension $1$. This gives a new proof of a theorem of Phong–Sturm.

Classification : 14E15, 32S45
Keywords: Resolution of singularities, log-canonical threshold, ascending chain condition
Mot clés : Résolution des singularités, seuil log-canonique, condition de chaîne ascendante

[1] Alexeev, Valery Two-dimensional terminations, Duke Math. J., Volume 69 (1993) no. 3, pp. 527-545 | Zbl

[2] Artal Bartolo, Enrique; Cassou-Noguès, Pierrette; Luengo, Ignacio; Melle-Hernández, Alejandro On the log-canonical threshold for germs of plane curves, Singularities I. Algebraic and analytic aspects (Contemporary Mathematics), Volume 474, American Mathematical Society, 2008, pp. 1-14 | Zbl

[3] Berman, Robert J. Kähler-Einstein metrics, canonical random point processes and birational geometry (2016) (https://arxiv.org/abs/1307.3634)

[4] Bierstone, Edward; Milman, Pierre D. Semianalytic and subanalytic sets, Publ. Math., Inst. Hautes Étud. Sci., Volume 67 (1988), pp. 5-42 | Zbl

[5] Bierstone, Edward; Milman, Pierre D. Arc-analytic functions, Invent. Math., Volume 101 (1990) no. 2, pp. 411-424 | Zbl

[6] Collins, Tristan C.; Greenleaf, Allan; Pramanik, Malabika A multi-dimensional resolution of singularities with applications to analysis, Am. J. Math., Volume 135 (2013) no. 5, pp. 1179-1252 | Zbl

[7] Demailly, Jean-Pierre; Kollár, János Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér., Volume 34 (2001) no. 4, pp. 525-556 | Zbl

[8] Denef, Jan; Loeser, François Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J. Am. Math. Soc., Volume 5 (1992) no. 4, pp. 705-720 | Zbl

[9] Favre, Charles; Jonsson, Mattias Valuations and multiplier ideals, J. Am. Math. Soc., Volume 18 (2005) no. 3, pp. 655-684 | Zbl

[10] de Fernex, Tommaso; Ein, Lawrence; Mustaţă, Mircea Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties, Duke Math. J., Volume 152 (2010) no. 1, pp. 93-114 | Zbl

[11] de Fernex, Tommaso; Mustaţă, Mircea Limits of log-canonical thresholds, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 3, pp. 493-517 | Zbl

[12] Greenblatt, Michael Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Math. Ann., Volume 346 (2010) no. 4, pp. 857-895 | Zbl

[13] Hacon, Christopher D.; McKernan, James; Xu, Chenyang ACC for log canonical thresholds, Ann. Math., Volume 180 (2014) no. 2, pp. 523-571 | Zbl

[14] Hai, Le Mau; Hiep, Pham Hoang; Hung, Vu Viet The log canonical threshold of holomorphic functions, Int. J. Math., Volume 23 (2012) no. 11, 1250115, 8 pages (Art. ID 1250115, 8 p.) | Zbl

[15] Igusa, Jun-Ichi On the first terms of certain asymptotic expansions, Complex analysis and algebraic geometry, Iwanami Shoten, 1977, pp. 357-368 | Zbl

[16] Flips and Abundance for Algebraic Threefolds (Kollár, János, ed.), Astérisque, 211, Société Mathématique de France, 1992, 258 pages | Zbl

[17] Kollár, János Singularities of pairs, Algebraic geometry (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 221-285 | Zbl

[18] Kollár, János Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, 2007, vi+208 pages | Zbl

[19] Kollár, János Which powers of holomorphic functions are integrable? (2008) (https://arxiv.org/abs/0805.0756)

[20] Kuwata, Takayasu On the log canonical thresholds of reducible plane curves, Am. J. Math., Volume 121 (1999) no. 4, pp. 701-721 | Zbl

[21] Lin, Shaowei Ideal-theoretic strategies for asymptotic approximation of marginal likelihood integrals, J. Algebra, Volume 8 (2017) no. 1, pp. 22-55 | Zbl

[22] MacLagan, Diane Antichains of monomial ideals are finite, Proc. Am. Math. Soc., Volume 129 (2001), pp. 1609-1615 | Zbl

[23] Parusiński, Adam Subanalytic functions, Trans. Am. Math. Soc., Volume 344 (1994) no. 2, pp. 583-595 | Zbl

[24] Parusiński, Adam On the preparation theorem for subanalytic functions, New developments in singularity theory (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 21, Kluwer Academic Publishers, 2001, pp. 193-215 | Zbl

[25] Phong, Duong Hong; Stein, Elias Menachem The Newton polyhedron and oscillatory integral operators, Acta Math., Volume 179 (1997) no. 1, pp. 105-152 | Zbl

[26] Phong, Duong Hong; Stein, Elias Menachem; Sturm, J. A. On the growth and stability of real-analytic functions, Am. J. Math., Volume 121 (1999) no. 3, pp. 519-554 | Zbl

[27] Phong, Duong Hong; Sturm, J. A. Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions, Ann. Math., Volume 152 (2000) no. 1, pp. 277-329 | Zbl

[28] Phong, Duong Hong; Sturm, J. A. On a conjecture of Demailly and Kollár, Asian J. Math., Volume 4 (2000) no. 1, pp. 221-226 | Zbl

[29] Shokurov, Vyacheslav V. Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 56 (1992), pp. 105-203

[30] Shokurov, Vyacheslav V. 3-fold log flips, Russ. Acad. Sci., Izv., Math., Volume 40 (1993) no. 1, pp. 95-202 | Zbl

[31] Siu, Yum-Tong The existence of Kähler-Einstein metrics on manifolds with positive anticanonical bundle and suitable finite symmetry group, Ann. Math., Volume 127 (1987) no. 3, pp. 585-627 | Zbl

[32] Tian, Gang On Kähler-Einstein metrics on certain manifolds with $C_1\left(M\right)>0$, Invent. Math., Volume 89 (1987), pp. 225-246 | Zbl

[33] Tian, Gang On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1992) no. 1, pp. 101-172 | Zbl

[34] Tian, Gang; Yau, Shing Tung Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (Advanced Series in Mathematical Physics), Volume 1, World Scientific, 1986, pp. 574-628 | Zbl

[35] Tian, Gang; Yau, Shing Tung Complete Kähler manifolds with zero Ricci curvature II, Invent. Math., Volume 106 (1991) no. 1, pp. 27-60 | Zbl

[36] Varcenko, A. N. Newton polyhedra and estimates of oscillatory integrals, Funkts. Anal. Prilozh., Volume 10 (1976) no. 3, pp. 13-38