Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Automorphism groups of Calabi-Yau manifolds of Picard number $ 2$


Author: Keiji Oguiso
Journal: J. Algebraic Geom. 23 (2014), 775-795
DOI: https://doi.org/10.1090/S1056-3911-2014-00642-1
Published electronically: April 29, 2014
MathSciNet review: 3263669
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Abstract | References | Additional Information

Abstract: We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number $ 2$ is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperkähler manifolds and birational automorphism groups, as we shall see. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperkähler manifold of Picard number $ 2$. We will also discuss a similar conjectual relation together with the exsistence of a rational curve, expected by the cone conjecture, for a Calabi-Yau threefold of Picard number $ 2$.


References [Enhancements On Off] (What's this?)

  • [BS09] S. Boissiere and A. Sarti, A note on automorphisms and birational transformations of holomorphic symplectic manifolds, to appear in Proc. AMS, arXiv:0905.4370.
  • [Bu05] W. Burnside, On criteria for the finiteness of the order of a group of linear substituions, Proc. London Math. Soc. S2-3, no. 1, 435. MR 1575943, https://doi.org/10.1112/plms/s2-3.1.435
  • [Ca07] Andrei Căldăraru, Non-birational Calabi-Yau threefolds that are derived equivalent, Internat. J. Math. 18 (2007), no. 5, 491-504. MR 2331075 (2008h:14017), https://doi.org/10.1142/S0129167X07004205
  • [CO11] S. Cantat and K. Oguiso, Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups, arXiv:1107.5862.
  • [DF11] S. Diverio and A. Ferretti, On a conjecture of Oguiso about rational curves on Calabi-Yau threefolds, arXiv:1107.3337, to appear in Comment. Math. Helv.
  • [GHJ03] M. Gross, D. Huybrechts, and D. Joyce, Calabi-Yau manifolds and related geometries, Universitext, Springer-Verlag, Berlin, 2003. Lectures from the Summer School held in Nordfjordeid, June 2001. MR 1963559 (2004c:14075)
  • [GP01] Mark Gross and Sorin Popescu, Calabi-Yau threefolds and moduli of abelian surfaces. I, Compositio Math. 127 (2001), no. 2, 169-228. MR 1845899 (2002f:14057), https://doi.org/10.1023/A:1012076503121
  • [Hu99] Daniel Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63-113. Erratum: ``Compact hyper-Kähler manifolds: basic results'', Invent. Math. 152 (2003) 209-212.. MR 1664696 (2000a:32039), https://doi.org/10.1007/s002220050280
  • [Hu11] D. Huybrechts, A global Torelli theorem for hyperkaehler manifolds (after Verbitsky), arXiv:1106.5573.
  • [HT01] B. Hassett and Y. Tschinkel, Rational curves on holomorphic symplectic fourfolds, Geom. Funct. Anal. 11 (2001), no. 6, 1201-1228. MR 1878319 (2002m:14033), https://doi.org/10.1007/s00039-001-8229-1
  • [HT09] Brendan Hassett and Yuri Tschinkel, Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal. 19 (2009), no. 4, 1065-1080. MR 2570315 (2011c:14119), https://doi.org/10.1007/s00039-009-0022-6
  • [HT10] Brendan Hassett and Yuri Tschinkel, Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces, J. Inst. Math. Jussieu 9 (2010), no. 1, 125-153. MR 2576800 (2011c:14044), https://doi.org/10.1017/S1474748009000140
  • [HW92] D. R. Heath-Brown and P. M. H. Wilson, Calabi-Yau threefolds with $ \rho >13$, Math. Ann. 294 (1992), no. 1, 49-57. MR 1180449 (93j:14046), https://doi.org/10.1007/BF01934312
  • [Ka88] Yujiro Kawamata, Crepant blowing-up of $ 3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93-163. MR 924674 (89d:14023), https://doi.org/10.2307/1971417
  • [Ka97] Yujiro Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665-687. MR 1468356 (98g:14043), https://doi.org/10.1142/S0129167X97000354
  • [Ka08] Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419-423. MR 2426353 (2009d:14011), https://doi.org/10.2977/prims/1210167332
  • [Ko87] Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698 (89e:53100)
  • [Ko94] Sándor J. Kovács, The cone of curves of a $ K3$ surface, Math. Ann. 300 (1994), no. 4, 681-691. MR 1314742 (96a:14044), https://doi.org/10.1007/BF01450509
  • [Ku03] Marco Kühnel, Calabi-Yau-threefolds with Picard number $ \rho (X)=2$ and their Kähler cone. I, Math. Z. 245 (2003), no. 2, 233-254. MR 2013500 (2005b:32057), https://doi.org/10.1007/s00209-003-0540-0
  • [Ku04] Marco Kühnel, Calabi-Yau threefolds with Picard number $ \rho (X)=2$ and their Kähler cone. II, Pacific J. Math. 217 (2004), no. 1, 115-137. MR 2105769 (2005i:14050), https://doi.org/10.2140/pjm.2004.217.115
  • [Ma11] Eyal Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp. 257-322. MR 2964480, https://doi.org/10.1007/978-3-642-20300-8_15
  • [Mo93] David R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243-271. Journées de Géométrie Algébrique d'Orsay (Orsay, 1992). MR 1265317 (95d:32021)
  • [Ogr05] K. G. O'Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal. 15 (2005), no. 6, 1223-1274. MR 2221247 (2007b:14025), https://doi.org/10.1007/s00039-005-0538-3
  • [Og93] Keiji Oguiso, On algebraic fiber space structures on a Calabi-Yau $ 3$-fold, Internat. J. Math. 4 (1993), no. 3, 439-465. With an appendix by Noboru Nakayama. MR 1228584 (94g:14019), https://doi.org/10.1142/S0129167X93000248
  • [OP98] Keiji Oguiso and Thomas Peternell, Calabi-Yau threefolds with positive second Chern class, Comm. Anal. Geom. 6 (1998), no. 1, 153-172. MR 1619841 (99c:14052)
  • [OS01] Keiji Oguiso and Jun Sakurai, Calabi-Yau threefolds of quotient type, Asian J. Math. 5 (2001), no. 1, 43-77. MR 1868164 (2002i:14043)
  • [PSS71] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli's theorem for algebraic surfaces of type $ {\rm K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530-572 (Russian). MR 0284440 (44 #1666)
  • [Sa07] Justin Sawon, Lagrangian fibrations on Hilbert schemes of points on $ K3$ surfaces, J. Algebraic Geom. 16 (2007), no. 3, 477-497. MR 2306277 (2008d:14008), https://doi.org/10.1090/S1056-3911-06-00453-X
  • [Sc11] C. Schnell, The fundamental group is not a derived invariant, arXiv:1112.3586
  • [Si91] Joseph H. Silverman, Rational points on $ K3$ surfaces: a new canonical height, Invent. Math. 105 (1991), no. 2, 347-373. MR 1115546 (92k:14025), https://doi.org/10.1007/BF01232270
  • [St85] Hans Sterk, Finiteness results for algebraic $ K3$ surfaces, Math. Z. 189 (1985), no. 4, 507-513. MR 786280 (86j:14038), https://doi.org/10.1007/BF01168156
  • [Ve09] M. Verbitsky, A global Torelli theorem for hyperkähler manifolds, arXiv:0908.4121.
  • [We88] Joachim Wehler, $ K3$-surfaces with Picard number $ 2$, Arch. Math. (Basel) 50 (1988), no. 1, 73-82. MR 925498 (89b:14054), https://doi.org/10.1007/BF01313498
  • [Wi89] P. M. H. Wilson, Calabi-Yau manifolds with large Picard number, Invent. Math. 98 (1989), no. 1, 139-155. MR 1010159 (90h:14044), https://doi.org/10.1007/BF01388848
  • [Wi92] P. M. H. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561-583. MR 1150602 (93a:14037), https://doi.org/10.1007/BF01231902
  • [Wi97] P. M. H. Wilson, The role of $ c_2$ in Calabi-Yau classification--a preliminary survey, Mirror symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1997, pp. 381-392. MR 1416342 (98g:14045)
  • [Yo01] Kōta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817-884. MR 1872531 (2002k:14020), https://doi.org/10.1007/s002080100255
  • [Yo12] K. Yoshioka, Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface, preprint.


Additional Information

Keiji Oguiso
Affiliation: Department of Mathematics, Osaka University, Toyonaka 560-0043 Osaka, Japan; and Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Korea
Email: oguiso@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-2014-00642-1
Received by editor(s): June 18, 2012
Received by editor(s) in revised form: March 6, 2013
Published electronically: April 29, 2014
Additional Notes: This work was supported by JSPS Gran-in-Aid (B) No. 22340009, JSPS Grant-in-Aid (S) No. 22224001, and by KIAS Scholar Program
Dedicated: Dedicated to Professor Yujiro Kawamata on the occasion of his sixtieth birthday
Article copyright: © Copyright 2014

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