Abstract
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in Di Cerbo and Di Cerbo (see [5]), to study the geometry of cusped complex hyperbolic surfaces and their compactifications.
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References
A. Ash et al. Smooth compactifications of locally symmetric varieties. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010.
Baily W.L., Borel A.: Compactifications of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84, 442–528 (1966)
W. Barth et al. Compact complex surfaces. Ergebnisse der Mathematik undihrer Grenzgebiete (3), 4. Springer-Verlag, Berlin 2004.
A. Borel and L. Ji, Compactifications of locally symmetric spaces. Mathematics: Theory & Applications, Birkäuser Boston, Inc. Boston, MA, 2006.
G. Di Cerbo and L. F. Di Cerbo, Effective results for complex hyperbolic manifolds, arXiv:1212.0501 [mathDG], 2012.
L. F. Di Cerbo, Classification of toroidal compactifications with \({3\overline{c}_{2}=\overline{c}^{2}_{1}}\) and \({\overline{c}_{2}=1}\), arXiv:1309.5516 [mathAG], 2013.
Di Cerbo L.F.: Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications. Pacific J. Math. 255, 305–315 (2012)
Fujiki A.: An L 2 Dolbeault lemma and its applications. Publ. Res. Inst. Math. Sci. 28, 845–884 (1992)
Griffiths P., Harris J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley-Interscience, New York (1978)
Gromov M.: Volume and bounded cohomology. Publ. Math. Inst. Hautes Et́udes Sci. 56, 5–99 (1982)
Hirzebruch F.: Chern numbers of algebraic surfaces: an example. Math. Ann. 266, 351–356 (1982)
Hwang J.-M.: On the number of complex hyperbolic manifolds of bounded volume. Internat. J. Math. 8, 863–873 (2005)
J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32. Springer-Verlag, Berlin, 1996.
Mumford D.: Hirzebruch’s proportionality theorem in the non-compact case. Invent. Math. 42, 239–272 (1977)
Parker J.R.: On the volume of cusped, complex hyperbolic manifolds and orbifolds. Duke Math. J. 94, 433–464 (1998)
Reider I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)
Stover M.: On the number of ends of rank one locally symmetric spaces. Geom. Topol. 17, 905–924 (2013)
Friedman R.: Algebraic Surfaces and Holomorphic Vector Bundles, Univesitext. Springer-Verlag, New York (1998)
Y. T. Siu and S. T. Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Seminars in Differential Geometry, pp. 363–380, Ann. of Math. Stud., Vol. 102, Princeton Univ. Press, Princeton, N. J., 1982.
H. C. Wang, Topics on totally discontinuous groups, Symmetric Spaces, 459–487, edited by W. B. Boothby and G. L. Weiss, Pure and Appl. Math. 8, Marcel Dekker, New York, 1972.
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Cerbo, G.D., Di Cerbo, L.F. A sharp cusp count for complex hyperbolic surfaces and related results. Arch. Math. 103, 75–84 (2014). https://doi.org/10.1007/s00013-014-0666-9
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DOI: https://doi.org/10.1007/s00013-014-0666-9