×

Small bound for birational automorphism groups of algebraic varieties (with an appendix by Yujiro Kawamata). (English) Zbl 1128.14029

Let \(X\) be a smooth projective variety of general type and \(\text{alb}_X:X\to \text{Alb}(X)\) be its Albanese morphism. In the article under review, it is shown that if \(X\) has dimension \(n\geq 3\), \(X\to \text{alb}_X(X)\) is generically finite, \(\text{alb}_X(X)\) is smooth and \(\text{Alb}(X)\) is simple, then the order of the birational automorphism group \(| \text{Bir} (X)| \) is bounded by \(d_nV^{10}\) where \(d_n\) is a constant depending only on \(n\) and \(V\) is the canonical volume of \(X\). A stronger (and optimum) bound \(| \text{Bir} (X)| \leq 42^3V/3\) is obtained when \(n=3\), \(q(X)\geq 4\) and the dimension of \(\text{alb}_X(X)\) is \(<3\).

MSC:

14J50 Automorphisms of surfaces and higher-dimensional varieties
14E07 Birational automorphisms, Cremona group and generalizations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramovich D. (1994). Subvarieties of semiabelian varieties. Compos. Math. 90(1): 37–52 · Zbl 0814.14041
[2] Angehrn U. and Siu Y. (1995). Effective freeness and point separation for adjoint bundles. Invent. Math. 122(2): 291–308 · Zbl 0847.32035 · doi:10.1007/BF01231446
[3] Aljadeff E. and Sonn J. (1998). Bounds on orders of finite subgroups of PGL n (K). J. Algebra 210(1): 352–360 · Zbl 0919.20029 · doi:10.1006/jabr.1998.7551
[4] Aschbacher M. (2004). The status of the classification of the finite simple groups. Notices Am. Math. Soc. 51(7): 736–740 · Zbl 1113.20302
[5] Birkenhake C. and Lange H. (2004). Complex Abelian Varieties, 2nd edn. Grundlehren der Mathematischen Wissenschaften, 302. Springer, Berlin · Zbl 1056.14063
[6] Bombieri E. (1973). Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. 42: 171–219 · Zbl 0259.14005 · doi:10.1007/BF02685880
[7] Cai J. (1995). On abelian automorphism groups of threefolds of general type. Algebra Colloq. 2(4): 373–382 · Zbl 0846.14004
[8] Catanese F. and Schneider M. (1995). Polynomial bounds for abelian groups of automorphisms. Spec. Issue Honour Frans Oort. Compos. Math. 97(1–2): 1–15 · Zbl 0858.14008
[9] Chen J.A. and Hacon C.D. (2002). Linear Series of Irregular Varieties. Algebraic Geometry in East Asia (Kyoto, 2001), 143–153. World Sci. Publishing, River Edge, NJ · Zbl 1094.14502
[10] Chen M. (2004). Inequalities of Noether type for threefolds of general type. J. Math. Soc. Jpn. 56(4): 1131–1155 · Zbl 1079.14046 · doi:10.2969/jmsj/1190905452
[11] Conway J.H., Curtis R.T., Norton S.P., Parker R.A. and Wilson R.A. (1985). Atlas of Finite Groups. Oxford University Press, Eynsham · Zbl 0568.20001
[12] Corti A. (1991). Polynomial bounds for the number of automorphisms of a surface of general type. Ann. Sci. École Norm. Sup. (4) 24(1): 113–137 · Zbl 0758.14026
[13] Demailly J.-P., Peternell T. and Schneider M. (1994). Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3(2): 295–345 · Zbl 0827.14027
[14] Friedland S. (1997). The maximal orders of finite subgroups in GL n (Q). Proc. Am. Math. Soc. 125(12): 3519–3526 · Zbl 0895.20039 · doi:10.1090/S0002-9939-97-04283-4
[15] Fujita T. (1978). On Kahler fiber spaces over curves. J. Math. Soc. Jpn. 30(4): 779–794 · Zbl 0393.14006 · doi:10.2969/jmsj/03040779
[16] Fulton W. (1984). Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer, Berlin · Zbl 0541.14005
[17] Gorenstein D. (1968). Finite groups. Harper & Row, Publishers, New York · Zbl 0185.05701
[18] Griffiths P. and Harris J. (1994). Principles of Algebraic Geometry. Wiley &, New York · Zbl 0836.14001
[19] Griffiths P. and Harris J. (1979). Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12(3): 355–452 · Zbl 0426.14019
[20] Hacon C.D. (2004). On the degree of the canonical maps of threefolds. Proc. Jpn. Acad. Ser. A Math. Sci. 80(8): 166–167 · Zbl 1068.14046 · doi:10.3792/pjaa.80.166
[21] Hacon, C.D., McKernan, J.: Boundedness of pluricanonical maps of varieties of general type, math.AG/0504327 · Zbl 1121.14011
[22] Hanamura M. (1990). The birational automorphism groups and the Albanese maps of varieties with Kodaira dimension zero. J. Reine Angew. Math. 411: 124–136 · Zbl 0717.14008 · doi:10.1515/crll.1990.411.124
[23] Hartshorne R. (1971). Ample vector bundles on curves. Nagoya Math. J. 43: 73–89 · Zbl 0218.14018
[24] Hartshorne R. (1977). Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York · Zbl 0367.14001
[25] Heier G. (2005). Effective finiteness theorems for maps between canonically polarized compact complex manifolds. Math. Nachr. 278(1–2): 133–140 · Zbl 1068.14069 · doi:10.1002/mana.200310230
[26] Howard, A., Sommese, A.J.: On the orders of the automorphism groups of certain projective manifolds. Manifolds and Lie groups (Notre Dame, Ind., 1980), pp. 145–158, Progr. Math., 14, Birkhauser, Boston (1981)
[27] Huckleberry A.T. and Sauer M. (1990). On the order of the automorphism group of a surface of general type. Math. Z. 205(2): 321–329 · Zbl 0718.32022 · doi:10.1007/BF02571245
[28] Kawamata Y. (1982). Kodaira dimension of algebraic fiber spaces over curves. Invent. Math. 66(1): 57–71 · Zbl 0477.14011 · doi:10.1007/BF01404756
[29] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam (1987)
[30] Kollar J. (1986). Higher direct images of dualizing sheaves. I. Ann. Math. (2) 123(1): 11–42 · Zbl 0598.14015 · doi:10.2307/1971351
[31] Kollar, J.: Singularities of pairs. Algebraic geometry–Santa Cruz 1995, 221–287, Proc. Sympos. Pure Math., 62, Part 1, Am. Math. Soc., Providence, RI (1997)
[32] Kollar, J., Mori, S.: Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens, A. Corti. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge (1998) · Zbl 0926.14003
[33] Kovacs, S.J.: Number of automorphisms of principally polarized abelian varieties. Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), 3–7, Contemp. Math., 276, Am. Math. Soc., Providence (2001)
[34] Lazarsfeld, R.: Positivity in algebraic geometry. I; II. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48; 49. Springer, Berlin (2004) · Zbl 1066.14021
[35] Lee S. (2000). Quartic-canonical systems on canonical threefolds of index 1. Spec. Issue Honor Robin Hartshorne. Comm. Algebra 28(12): 5517–5530 · Zbl 1083.14518
[36] Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Sendai, 1985, 449–476, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam (1987)
[37] Mori, S.: Classification of higher-dimensional varieties. Algebraic geometry, Bowdoin, 1985, 269–331, Proc. Sympos. Pure Math., 46, Part 1, Am. Math. Soc., Providence (1987)
[38] Ran Z. (1984). The structure of Gauss-like maps. Compos. Math. 52(2): 171–177 · Zbl 0547.14004
[39] Reid, M.: Young person’s guide to canonical singularities. Algebraic geometry, Bowdoin, 1985, 345–414, Proc. Sympos. Pure Math., 46, Part 1, Am. Math. Soc., Providence (1987)
[40] Szabo E. (1996). Bounding automorphism groups. Math. Ann. 304(4): 801–811 · Zbl 0845.14009 · doi:10.1007/BF01446320
[41] Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics, Vol. 439. Springer, Berlin (1975) · Zbl 0299.14007
[42] Ueno K. (1976). A remark on automorphisms of Enriques surfaces. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 23(1): 149–165 · Zbl 0332.14014
[43] Viehweg E. (1982). Die Additivitat der Kodaira Dimension fur projektive Faserraume uber Varietaten des allgemeinen Typs. J. Reine Angew. Math. 330: 132–142 · Zbl 0466.14009 · doi:10.1515/crll.1982.330.132
[44] Weisfeiler, B.: Post-classification version of Jordan’s theorem on finite linear groups. Proc. Nat. Acad. Sci. U.S.A. 81, no. 16, Phys. Sci., pp. 5278–5279 (1984) · Zbl 0542.20026
[45] Xiao G. (1990). On abelian automorphism group of a surface of general type. Invent. Math. 102(3): 619–631 · Zbl 0739.14024 · doi:10.1007/BF01233441
[46] Xiao G. (1994). Bound of automorphisms of surfaces of general type. I. Ann. of Math. (2) 139(1): 51–77 · Zbl 0811.14011 · doi:10.2307/2946627
[47] Xiao G. (1995). Bound of automorphisms of surfaces of general type. II. J. Algebra Geom. 4(4): 701–793 · Zbl 0841.14011
[48] Xiao G. (1996). Linear bound for abelian automorphisms of varieties of general type. J. Reine Angew. Math. 476: 201–207 · Zbl 0870.14006 · doi:10.1515/crll.1996.476.201
[49] Yau S.T. (1978). On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampére equation. I. Comm. Pure Appl. Math. 31(3): 339–411 · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.