×
Hacon, Christopher D.; McKernan, James; Xu, Chenyang

On the birational automorphisms of varieties of general type. (English) Zbl 1281.14036

Ann. Math. (2) 177, No. 3, 1077-1111 (2013).
It is known that varieties of general type have a finite automorphism group [H. Matsumura, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 151–155 (1963; Zbl 0134.16601)]. A celebrated result of Hurwitz bounds the order of the automorphism group of a curve \(C\) of general type by \(84(g(C)-1)\). More recently G. Xiao [J. Algebr. Geom. 4, No. 4, 701–793 (1995; Zbl 0841.14011)] proved an analog of Hurwitz theorem for surfaces of general type. Note that the genus of a smooth curve can be interpreted in terms of volume of the canonical class. It is therefore natural to ask if it is possible to bound the order of the automorpshism group of a variety of general type, say \(X\), with a constant depending only on the dimension and \(\mathrm{vol}(X,K_X)\). This impressive paper answers completely to this question and proves the excistence of a universal constant c, depending only on the dimension, such that for any projective variety of general type \(X\) the dimension of the birational automorphism group has at most \(c\cdot \mathrm{vol}(X,K_X)\) elements. This generalization is highly non trivial and uses all the tools and techniques of Minimal Model Program, in particular there is a very subtle part on Descending Chain Conditions, that in the authors words could provide an affirmative answer to the more general Kollár’s DCC conjectures [J. Kollár, Contemp. Math. 162, 261–275 (1994; Zbl 0860.14014)]. The paper is very well written.
Reviewer: Massimiliano Mella (Ferrara)

Cited in 2 Reviews
Cited in 36 Documents

MathOverflow Questions:

Can the automorphism group vary too much in families of complex projective varieties?
Reference request: birational automorphism group is finite

MSC:

14J50 Automorphisms of surfaces and higher-dimensional varieties
14E30 Minimal model program (Mori theory, extremal rays)
14B05 Singularities in algebraic geometry

Keywords:

automorphism; varieties of general type; volume

Citations:

Zbl 0134.16601; Zbl 0841.14011; Zbl 0860.14014
PDF BibTeX XML Cite
\textit{C. D. Hacon} et al., Ann. Math. (2) 177, No. 3, 1077--1111 (2013; Zbl 1281.14036)
Full Text: DOI arXiv

References:

[1] V. Alexeev, ”Boundedness and \(K^2\) for log surfaces,” Internat. J. Math., vol. 5, iss. 6, pp. 779-810, 1994. · Zbl 0838.14028
[2] V. Alexeev and S. Mori, ”Bounding singular surfaces of general type,” in Algebra, Arithmetic and Geometry with Applications, New York: Springer-Verlag, 2004, pp. 143-174. · Zbl 1103.14021
[3] A. Andreotti, ”Sopra le superficie algebriche che posseggono trasformazioni birazionali in sè,” Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl., vol. 9, pp. 255-279, 1950. · Zbl 0041.08502
[4] U. Angehrn and Y. T. Siu, ”Effective freeness and point separation for adjoint bundles,” Invent. Math., vol. 122, iss. 2, pp. 291-308, 1995. · Zbl 0847.32035
[5] E. Ballico, ”On the automorphisms of surfaces of general type in positive characteristic,” Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., vol. 4, iss. 2, pp. 121-129, 1993. · Zbl 0795.14022
[6] B. Berndtsson and M. Puaun, ”Quantitative extensions of pluricanonical forms and closed positive currents,” Nagoya Math. J., vol. 205, pp. 25-65, 2012. · Zbl 1248.32012
[7] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, ”Existence of minimal models for varieties of log general type,” J. Amer. Math. Soc., vol. 23, iss. 2, pp. 405-468, 2010. · Zbl 1210.14019
[8] J. Cai, ”Bounds of automorphisms of surfaces of general type in positive characteristic,” J. Pure Appl. Algebra, vol. 149, iss. 3, pp. 241-250, 2000. · Zbl 0968.14023
[9] F. Catanese and M. Schneider, ”Polynomial bounds for abelian groups of automorphisms,” Compositio Math., vol. 97, iss. 1-2, pp. 1-15, 1995. · Zbl 0858.14008
[10] A. Corti, ”Polynomial bounds for the number of automorphisms of a surface of general type,” Ann. Sci. École Norm. Sup., vol. 24, iss. 1, pp. 113-137, 1991. · Zbl 0758.14026
[11] T. de Fernex and C. D. Hacon, ”Deformations of canonical pairs and Fano varieties,” J. Reine Angew. Math., vol. 651, pp. 97-126, 2011. · Zbl 1220.14026
[12] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993, vol. 131. · Zbl 0813.14039
[13] D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Providence, RI: Amer. Math. Soc., 1994, vol. 40. · Zbl 0816.20016
[14] A. Grothendieck, ”Eléménts de Géométrie Algébrique. IV,” Publ. Math. IHES, vol. 28, pp. 5-255, 1966. · Zbl 0144.19904
[15] C. D. Hacon and J. McKernan, ”Boundedness of pluricanonical maps of varieties of general type,” Invent. Math., vol. 166, iss. 1, pp. 1-25, 2006. · Zbl 1121.14011
[16] A. T. Huckleberry and M. Sauer, ”On the order of the automorphism group of a surface of general type,” Math. Z., vol. 205, iss. 2, pp. 321-329, 1990. · Zbl 0718.32022
[17] J. Kollár, ”Log surfaces of general type; some conjectures,” in Classification of Algebraic Varieties, Providence, RI: Amer. Math. Soc., 1994, vol. 162, pp. 261-275. · Zbl 0860.14014
[18] J. Kollár, ”Singularities of pairs,” in Algebraic Geometry-Santa Cruz 1995, Providence, RI: Amer. Math. Soc., 1997, vol. 62, pp. 221-287. · Zbl 0905.14002
[19] J. Kollár, ”Is there a topological Bogomolov-Miyaoka-Yau inequality?,” Pure Appl. Math. Q., vol. 4, pp. 203-236, 2008. · Zbl 1145.14031
[20] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134. · Zbl 0926.14003
[21] R. Lazarsfeld, Positivity in Algebraic Geometry. II, New York: Springer-Verlag, 2004, vol. 49. · Zbl 1093.14500
[22] H. Matsumura, ”On algebraic groups of birational transformations,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 34, pp. 151-155, 1963. · Zbl 0134.16601
[23] N. Nakayama, Zariski-Decomposition and Abundance, Tokyo: Mathematical Society of Japan, 2004, vol. 14. · Zbl 1061.14018
[24] Y. Siu, ”Invariance of plurigenera,” Invent. Math., vol. 134, iss. 3, pp. 661-673, 1998. · Zbl 0955.32017
[25] H. Stichtenoth, ”Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe,” Arch. Math. \((\)Basel\()\), vol. 24, pp. 527-544, 1973. · Zbl 0282.14007
[26] E. Szabó, ”Bounding automorphism groups,” Math. Ann., vol. 304, iss. 4, pp. 801-811, 1996. · Zbl 0845.14009
[27] S. Takayama, ”Pluricanonical systems on algebraic varieties of general type,” Invent. Math., vol. 165, iss. 3, pp. 551-587, 2006. · Zbl 1108.14031
[28] H. Tsuji, Bound of automorphisms of projective varieties of general type. · Zbl 1186.14043
[29] H. Tsuji, ”Pluricanonical systems of projective varieties of general type. II,” Osaka J. Math., vol. 44, iss. 3, pp. 723-764, 2007. · Zbl 1186.14043
[30] G. Xiao, ”Bound of automorphisms of surfaces of general type. I,” Ann. of Math., vol. 139, iss. 1, pp. 51-77, 1994. · Zbl 0811.14011
[31] G. Xiao, ”Bound of automorphisms of surfaces of general type. II,” J. Algebraic Geom., vol. 4, iss. 4, pp. 701-793, 1995. · Zbl 0841.14011
[32] G. Xiao, ”Linear bound for abelian automorphisms of varieties of general type,” J. Reine Angew. Math., vol. 476, pp. 201-207, 1996. · Zbl 0870.14006
[33] D. Zhang, ”Small bound for birational automorphism groups of algebraic varieties,” Math. Ann., vol. 339, iss. 4, pp. 957-975, 2007. · Zbl 1128.14029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.