Abstract
We study complex surfaces with locally CAT(0) polyhedral Kähler metrics and construct such metrics on \({\mathbb{C}P^{2}}\) with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of \({\mathbb{C}P^{2}}\) of sufficiently high degree their desingularizations are of type K(π, 1).
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References
Alexander S., Bishop R.L.: Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above. Differential Geom. Appl. 6(1), 67–86 (1996)
S. Alexander, Kapovitch V. , Petrunin A. Alexandrov geometry, to appear.
B.H. Bowditch, Notes on locally CAT(1) spaces, Geometric Group Theory (R. Charney, M. Davis, M. Shapiro, eds.), de Gruyter (1995), 1–48.
Boyer C.P., Galicki K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)
Charney R., Davis M.: Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds. American Journal of Mathematics 115(5), 929–1009 (1993)
Charney R., Davis M., Moussong G.: Nonpositively curved piecewise Euclidean structures on hyperbolic manifolds. Michigan Math. J. 44, 201–208 (1997)
Couwenberg W., Heckman G.: Looijenga E. Geometric structures on the complement of a projective arrangement. Publ. Math. de l’IHES 101, 69–161 (2005)
Davis M., G. Moussong, Notes on nonpositively curved polyhedra, in Low Dimensional Topology, Bolyai Society Math. Studies 8, Janos Bolyai Math. Soc., Budapest (1999), 11–94
E. Ghys, P. de la Harpe, eds. Sur les groupes hyperboliques d’apres Mikhael Gromov, Progress in Mathematics. 83 (1990).
B. Grünbaum, Arrangements of hyperplanes, Proceedings of the Second Louisiana Conference on Combinatorics Graph Theory and Computings (1971) 41–106.
Gromov M.: Hyperbolic groups, Essays in Group Theory (S. Gersten, ed.). MSRI Publications Springer 8, 75–265 (1987)
M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory 2, Proc. Symp. Sussex Univ., Brighton, July 14-19 (1991) Lond. Math. Soc. Lecture Notes 182, (Niblo and Roller ed.), Cambridge Univ. Press, Cambridge (1993), 1–295.
Hempel J.: Residual finiteness of surface groups. Proc. Amer. Math. Soc. 32, 323 (1972)
Hirzebruch F.: Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II. Birkhäuser Progr. Math. 36, 113–140 (1983)
R. Kobayashi, S. Nakamura, F. Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. 65 Ser. A (1989) 238–241.
Mostow G.D., Siu Y.T.: A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. 112, 321–360 (1980)
Panov D.: Polyhedral Kähler manifolds. Geometry and Topology 13, 2205–2252 (2009)
Thurston W.P.: Shapes of polyhedra and triangulations of the sphere. Geometry and Topology Monographs 1, 511–549 (1998)
Toponogov V.A.: Evaluation of the length of a closed geodesic on a convex surface (in Russian). Dokl. Akad. Nauk SSSR 124, 282–284 (1959)
Uludag A.M.: Covering relations between ball-quotient orbifolds. Math. Ann. 328(3), 503–523 (2004)
Yoshida M.: Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory, Aspects of Mathematics, E11, Friedr. Vieweg and Sohn, Braunschweig (1987)
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Supported by a Royal Society University Research Fellowship.
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Panov, D. Complex Surfaces with Cat(0) Metrics. Geom. Funct. Anal. 21, 1218–1238 (2011). https://doi.org/10.1007/s00039-011-0133-8
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DOI: https://doi.org/10.1007/s00039-011-0133-8