On the Cartwright-Steger surface
Authors:
Donald I. Cartwright, Vincent Koziarz and Sai-Kee Yeung
Journal:
J. Algebraic Geom. 26 (2017), 655-689
DOI:
https://doi.org/10.1090/jag/696
Published electronically:
April 21, 2017
MathSciNet review:
3683423
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface, the unique smooth surface of Euler number 3 which is neither a projective plane nor a fake projective plane. In particular, we determine the genus of a generic fiber of the Albanese fibration and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface.
References
- M. A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968), 299–301. MR 221488, DOI https://doi.org/10.1017/s0305004100042845
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
- Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314
- Ragnar-Olaf Buchweitz and Gert-Martin Greuel, The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), no. 3, 241–281. MR 571575, DOI https://doi.org/10.1007/BF01390254
- Huai Dong Cao and Ngaiming Mok, Holomorphic immersions between compact hyperbolic space forms, Invent. Math. 100 (1990), no. 1, 49–61. MR 1037142, DOI https://doi.org/10.1007/BF01231180
- D. Cartwright, V. Koziarz, and S.-K. Yeung, On the Cartwright-Steger surface, long version, arXiv:1412.4137. Magma files are available on the HAL archive under the reference hal-01429836. See also http://www.maths. usyd.edu.au/u/donaldc/cs-surface/ or https://www.math.u-bordeaux.fr/ $\sim$vkoziarz/publi.html.
- Donald I. Cartwright and Tim Steger, Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 11–13 (English, with English and French summaries). MR 2586735, DOI https://doi.org/10.1016/j.crma.2009.11.016
- D. Cartwright and T. Steger, Finding generators and relations for groups acting on the hyperbolic ball, arXiv:1701.02452.
- P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
- Pierre Deligne and G. Daniel Mostow, Commensurabilities among lattices in ${\rm PU}(1,n)$, Annals of Mathematics Studies, vol. 132, Princeton University Press, Princeton, NJ, 1993. MR 1241644
- Martin Deraux, A negatively curved Kähler threefold not covered by the ball, Invent. Math. 160 (2005), no. 3, 501–525. MR 2178701, DOI https://doi.org/10.1007/s00222-004-0414-z
- Martin Deraux, Forgetful maps between Deligne-Mostow ball quotients, Geom. Dedicata 150 (2011), 377–389. MR 2753711, DOI https://doi.org/10.1007/s10711-010-9511-x
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- Jonghae Keum, Toward a geometric construction of fake projective planes, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23 (2012), no. 2, 137–155. MR 2924897, DOI https://doi.org/10.4171/RLM/622
- Ron Aharon Livne, ON CERTAIN COVERS OF THE UNIVERSAL ELLIPTIC CURVE, ProQuest LLC, Ann Arbor, MI, 1981. Thesis (Ph.D.)–Harvard University. MR 2936887
- G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
- G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106. MR 849652
- John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1–42. MR 2581913, DOI https://doi.org/10.1090/conm/501/09838
- Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321–370. MR 2289867, DOI https://doi.org/10.1007/s00222-007-0034-5
- Bertrand Rémy, Covolume des groupes $S$-arithmétiques et faux plans projectifs [d’après Mumford, Prasad, Klingler, Yeung, Prasad-Yeung], Astérisque 326 (2009), Exp. No. 984, vii, 83–129 (2010) (French, with French summary). Séminaire Bourbaki. Vol. 2007/2008. MR 2605319
- John Kurt Sauter Jr., Isomorphisms among monodromy groups and applications to lattices in ${\rm PU}(1,2)$, Pacific J. Math. 146 (1990), no. 2, 331–384. MR 1078386
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI https://doi.org/10.4153/cjm-1954-028-3
- Domingo Toledo, Maps between complex hyperbolic surfaces, Geom. Dedicata 97 (2003), 115–128. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999). MR 2003694, DOI https://doi.org/10.1023/A%3A1023691505890
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
- Gang Xiao, Surfaces fibrées en courbes de genre deux, Lecture Notes in Mathematics, vol. 1137, Springer-Verlag, Berlin, 1985 (French). MR 872271
- Sai-Kee Yeung, Classification of fake projective planes, Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), vol. 13, Int. Press, Somerville, MA, 2010, pp. 391–431. MR 2761486
- Sai-Kee Yeung, Classification of surfaces of general type with Euler number 3, J. Reine Angew. Math. 679 (2013), 1–22. MR 3065152, DOI https://doi.org/10.1515/CRELLE.2011.178
References
- M. A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968), 299–301. MR 0221488
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
- Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314
- Ragnar-Olaf Buchweitz and Gert-Martin Greuel, The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), no. 3, 241–281. MR 571575, DOI https://doi.org/10.1007/BF01390254
- Huai Dong Cao and Ngaiming Mok, Holomorphic immersions between compact hyperbolic space forms, Invent. Math. 100 (1990), no. 1, 49–61. MR 1037142, DOI https://doi.org/10.1007/BF01231180
- D. Cartwright, V. Koziarz, and S.-K. Yeung, On the Cartwright-Steger surface, long version, arXiv:1412.4137. Magma files are available on the HAL archive under the reference hal-01429836. See also http://www.maths. usyd.edu.au/u/donaldc/cs-surface/ or https://www.math.u-bordeaux.fr/ $\sim$vkoziarz/publi.html.
- Donald I. Cartwright and Tim Steger, Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 11–13 (English, with English and French summaries). MR 2586735, DOI https://doi.org/10.1016/j.crma.2009.11.016
- D. Cartwright and T. Steger, Finding generators and relations for groups acting on the hyperbolic ball, arXiv:1701.02452.
- P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
- Pierre Deligne and G. Daniel Mostow, Commensurabilities among lattices in $\textrm {PU}(1,n)$, Annals of Mathematics Studies, vol. 132, Princeton University Press, Princeton, NJ, 1993. MR 1241644
- Martin Deraux, A negatively curved Kähler threefold not covered by the ball, Invent. Math. 160 (2005), no. 3, 501–525. MR 2178701, DOI https://doi.org/10.1007/s00222-004-0414-z
- Martin Deraux, Forgetful maps between Deligne-Mostow ball quotients, Geom. Dedicata 150 (2011), 377–389. MR 2753711, DOI https://doi.org/10.1007/s10711-010-9511-x
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Jonghae Keum, Toward a geometric construction of fake projective planes, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23 (2012), no. 2, 137–155. MR 2924897, DOI https://doi.org/10.4171/RLM/622
- Ron Aharon Livne, On certain covers of the universal elliptic curve, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Harvard University, 1981. MR 2936887
- G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
- G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106. MR 849652
- John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1–42. MR 2581913, DOI https://doi.org/10.1090/conm/501/09838
- Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321–370. MR 2289867, DOI https://doi.org/10.1007/s00222-007-0034-5
- Bertrand Rémy, Covolume des groupes $S$-arithmétiques et faux plans projectifs [d’après Mumford, Prasad, Klingler, Yeung, Prasad-Yeung], Séminaire Bourbaki. Vol. 2007/2008, Astérisque 326 (2009), Exp. No. 984, vii, 83–129 (2010) (French, with French summary). MR 2605319
- John Kurt Sauter Jr., Isomorphisms among monodromy groups and applications to lattices in $\textrm {PU}(1,2)$, Pacific J. Math. 146 (1990), no. 2, 331–384. MR 1078386
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304. MR 0059914
- Domingo Toledo, Maps between complex hyperbolic surfaces, Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999), Geom. Dedicata 97 (2003), 115–128. MR 2003694, DOI https://doi.org/10.1023/A%3A1023691505890
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Notes written in collaboration with P. Cherenack, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. MR 0506253
- Gang Xiao, Surfaces fibrées en courbes de genre deux, Lecture Notes in Mathematics, vol. 1137, Springer-Verlag, Berlin, 1985 (French). MR 872271
- Sai-Kee Yeung, Classification of fake projective planes, Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), vol. 13, Int. Press, Somerville, MA, 2010, pp. 391–431. MR 2761486
- Sai-Kee Yeung, Classification of surfaces of general type with Euler number 3, J. Reine Angew. Math. 679 (2013), 1–22. Corrected version, http://www.math.purdue.edu/$\sim$yeung/. MR 3065152, DOI https://doi.org/10.1515/CRELLE.2011.178
Additional Information
Donald I. Cartwright
Affiliation:
School of Mathematics & Statistics, University of Sydney, Sydney, NSW 2006, Australia
MR Author ID:
45810
Email:
donald.cartwright@sydney.edu.au
Vincent Koziarz
Affiliation:
Institut de Mathématiques, Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France
MR Author ID:
638560
Email:
vincent.koziarz@math.u-bordeaux.fr
Sai-Kee Yeung
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID:
263917
Email:
yeung@math.purdue.edu
Received by editor(s):
June 4, 2015
Received by editor(s) in revised form:
December 29, 2015, and October 13, 2016
Published electronically:
April 21, 2017
Additional Notes:
The third author was partially supported by a grant from the National Science Foundation
Article copyright:
© Copyright 2017
University Press, Inc.
