Geometry & Topology

Classification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{2} = 3$

Luca F Di Cerbo and Matthew Stover

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We classify the minimum-volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces, and they are all arithmetic; ie they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an abelian surface at one point. The others are bielliptic surfaces blown up at one point. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.

Article information

Source
Geom. Topol., Volume 22, Number 4 (2018), 2465-2510.

Dates
Received: 6 February 2017
Revised: 20 June 2017
Accepted: 20 July 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1523584828

Digital Object Identifier
doi:10.2140/gt.2018.22.2465

Mathematical Reviews number (MathSciNet)
MR3784527

Zentralblatt MATH identifier
06864343

Subjects
Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 57M50: Geometric structures on low-dimensional manifolds
Secondary: 14M27: Compactifications; symmetric and spherical varieties 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds 11F60: Hecke-Petersson operators, differential operators (several variables)

Keywords
toroidal compactifications complex hyperbolic manifolds minimal volume manifolds arithmetic lattices

Citation

Di Cerbo, Luca F; Stover, Matthew. Classification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{2} = 3$. Geom. Topol. 22 (2018), no. 4, 2465--2510. doi:10.2140/gt.2018.22.2465. https://projecteuclid.org/euclid.gt/1523584828


Export citation

References

  • A Ash, D Mumford, M Rapoport, Y-S Tai, Smooth compactifications of locally symmetric varieties, 2nd edition, Cambridge Univ. Press (2010)
  • G Bagnera, M de Franchis, Sur les surfaces hyperelliptiques, C. R. Acad. Sci. Paris 145 (1908) 747–749
  • W P Barth, K Hulek, C A M Peters, A Van de Ven, Compact complex surfaces, 2nd edition, Ergeb. Math. Grenzgeb. 4, Springer (2004)
  • I Bauer, F Catanese, R Pignatelli, Surfaces of general type with geometric genus zero: a survey, from “Complex and differential geometry” (W Ebeling, K Hulek, K Smoczyk, editors), Springer Proc. Math. 8, Springer (2011) 1–48
  • A Beauville, Complex algebraic surfaces, 2nd edition, London Mathematical Society Student Texts 34, Cambridge Univ. Press (1996)
  • M Belolipetsky, Hyperbolic orbifolds of small volume, from “Proceedings of the International Congress of Mathematicians, II: Invited lectures” (S Y Jang, Y R Kim, D-W Lee, I Yie, editors), Kyung Moon Sa, Seoul (2014) 837–851
  • A Borel, L Ji, Compactifications of symmetric and locally symmetric spaces, Progress in Mathematics 229, Birkhäuser, Boston (2006)
  • W Bosma, J Cannon, C Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997) 235–265
  • D I Cartwright, T Steger, Enumeration of the $50$ fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010) 11–13
  • O Debarre, Tores et variétés abéliennes complexes, Cours Spécialisés 6, Soc. Math. France, Paris (1999)
  • P Deligne, G D Mostow, Commensurabilities among lattices in ${\rm PU}(1,n)$, Annals of Mathematics Studies 132, Princeton Univ. Press (1993)
  • G Di Cerbo, L F Di Cerbo, Effective results for complex hyperbolic manifolds, J. Lond. Math. Soc. 91 (2015) 89–104
  • L F Di Cerbo, Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications, Pacific J. Math. 255 (2012) 305–315
  • L F Di Cerbo, On the classification of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}$ and $\bar{c}_{2}=1$, preprint (2014)
  • L F Di Cerbo, M Stover, Multiple realizations of varieties as ball quotient compactifications, Michigan Math. J. 65 (2016) 441–447
  • L F Di Cerbo, M Stover, Bielliptic ball quotient compactifications and lattices in $\rm PU(2,1)$ with finitely generated commutator subgroup, Ann. Inst. Fourier $($Grenoble$)$ 67 (2017) 315–328
  • E Falbel, J R Parker, The geometry of the Eisenstein–Picard modular group, Duke Math. J. 131 (2006) 249–289
  • R Friedman, Algebraic surfaces and holomorphic vector bundles, Springer (1998)
  • D Gabai, R Meyerhoff, P Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009) 1157–1215
  • M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982) 5–99
  • R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer (1977)
  • F Hirzebruch, Chern numbers of algebraic surfaces: an example, Math. Ann. 266 (1984) 351–356
  • R-P Holzapfel, Chern numbers of algebraic surfaces–-Hirzebruch's examples are Picard modular surfaces, Math. Nachr. 126 (1986) 255–273
  • R-P Holzapfel, Ball and surface arithmetics, Aspects of Mathematics 29, Vieweg, Braunschweig (1998)
  • R-P Holzapfel, Complex hyperbolic surfaces of abelian type, Serdica Math. J. 30 (2004) 207–238
  • C Hummel, Rank one lattices whose parabolic isometries have no rotational part, Proc. Amer. Math. Soc. 126 (1998) 2453–2458
  • Y Kawamata, On deformations of compactifiable complex manifolds, Math. Ann. 235 (1978) 247–265
  • B Klingler, Sur la rigidité de certains groupes fondamentaux, l'arithméticité des réseaux hyperboliques complexes, et les “faux plans projectifs”, Invent. Math. 153 (2003) 105–143
  • Y Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977) 225–237
  • N Mok, Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, from “Perspectives in analysis, geometry, and topology” (I Itenberg, B Jöricke, M Passare, editors), Progr. Math. 296, Springer (2012) 331–354
  • A Momot, Irregular ball-quotient surfaces with non-positive Kodaira dimension, Math. Res. Lett. 15 (2008) 1187–1195
  • D Mumford, Hirzebruch's proportionality theorem in the noncompact case, Invent. Math. 42 (1977) 239–272
  • D Mumford, An algebraic surface with $K$ ample, $(K\sp{2})=9$, $p\sb{g}=q=0$, Amer. J. Math. 101 (1979) 233–244
  • G Prasad, S-K Yeung, Fake projective planes, Invent. Math. 168 (2007) 321–370
  • F Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980) 89–120
  • F Serrano, Divisors of bielliptic surfaces and embeddings in ${\bf P}^4$, Math. Z. 203 (1990) 527–533
  • M Stover, Volumes of Picard modular surfaces, Proc. Amer. Math. Soc. 139 (2011) 3045–3056
  • G Tian, S-T Yau, Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, from “Mathematical aspects of string theory” (S-T Yau, editor), Adv. Ser. Math. Phys. 1, World Scientific, Singapore (1987) 574–628
  • G Urzúa, Arrangements of curves and algebraic surfaces, J. Algebraic Geom. 19 (2010) 335–365
  • S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339–411
  • S-K Yeung, Integrality and arithmeticity of co-compact lattice corresponding to certain complex two-ball quotients of Picard number one, Asian J. Math. 8 (2004) 107–129