Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The maximum number of singular points on rational homology projective planes


Authors: Dongseon Hwang and Jonghae Keum
Journal: J. Algebraic Geom. 20 (2011), 495-523
DOI: https://doi.org/10.1090/S1056-3911-10-00532-1
Published electronically: March 24, 2010
MathSciNet review: 2786664
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Abstract | References | Additional Information

Abstract: A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $ \mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most $ 5$ singular points. So far all known examples have at most $ 4$ singular points. In this paper, we prove that a rational homology projective plane $ S$ with quotient singularities such that $ K_S$ is nef has at most $ 4$ singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type $ 3A_1 \oplus 2A_3$.

We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case.


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

  • 1. G. N. Belousov, Del Pezzo surfaces with log-terminal singularities, Mat. Zametki 83 (2008), no. 2, 170-180. MR 2431579
  • 2. D. Bindschadler and L. Brenton, On singular 4-manifolds of the homology type of $ CP^2$, J. Math. Kyoto Univ. 24 (1984), 67-81. MR 737825 (86i:32051)
  • 3. L. Brenton, D. Drucker and G. C. E. Prins, Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of $ CP^2$, Comment. Math. Helvetici 56 (1981), 39-58. MR 615614 (84e:32026b)
  • 4. L. Brenton, Some examples of singular compact analytic surfaces which are homotopy equivalent to the complex projective plane, Topology 16 (1977), 423-433. MR 0470253 (57:10011)
  • 5. E. Brieskorn, Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1968), 336-358. MR 0222084 (36:5136)
  • 6. S. Keel and J. McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669. MR 1610249 (99m:14068)
  • 7. J. Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), 919-927. MR 2239523 (2008b:14065)
  • 8. J. Keum, A rationality criterion for projective surfaces - partial solution to Kollar's conjecture, Algebraic geometry, 75-87, Contemp. Math. 422, Amer. Math. Soc., Providence, RI, 2007. MR 2296433 (2008a:14052)
  • 9. J. Keum, Quotients of fake projective planes, Geometry & Topology 12 (2008), 2497-2515. MR 2443971
  • 10. J. Keum, and D.-Q. Zhang, Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds, Jour. Pure Applied Algebra 170 (2002), 67-91. MR 1896342 (2002m:14029)
  • 11. R. Kobayashi, S. Nakamura, and F. Sakai A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A, Math. Sci. 65 (1989), no. 7, 238-241. MR 1030189 (90k:32034)
  • 12. J. Kollár, Einstein metrics on $ 5$-dimensional Seifert bundles, Jour. Geom. Anal.15 (2005), no. 3, 463-495. MR 2190241 (2007c:53056)
  • 13. J. Kollár, Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl. Math. Q. 4 (2008), no. 2, part 1, 203-236. MR 2400877 (2009b:14086)
  • 14. J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299-338. MR 922803 (88m:14022)
  • 15. S. Kondō, Enriques surfaces with finite automorphism groups, Japanese J. Math. 12 (1986), 191-282. MR 914299 (89c:14058)
  • 16. E. Looijenga, and J. Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986), no.3, 261-291. MR 842425 (87j:32032)
  • 17. M. Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89-118. MR 1116920 (92f:14028)
  • 18. K. Matsuki, Introduction to the Mori program, Universitext. Springer-Verlag, New York, 2002. MR 1875410 (2002m:14011)
  • 19. G. Megyesi, Generalisation of the Bogomolov-Miyaoka-Yau inequality to singular surfaces, Proc. London Math. Soc. (2) 78 (1999), 241-282. MR 1665244 (2000j:14058)
  • 20. Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159-171. MR 744605 (85j:14060)
  • 21. D. Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proc. of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, Springer, 1972, pp. 41-101. Lecture Notes in Math., Vol. 298. MR 0362383 (50:14825)
  • 22. V. V. Nikulin, Integral symmetric bilinear forms and its applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111-177; English translation: Math. USSR Izv. 14 (1979), no. 1, 103-167 (1980). MR 525944 (80j:10031)
  • 23. O. Riemenschneider, Die Invarianten der endlichen Untergruppen von GL(2, $ \mathbb{C}$), Math. Zeit. 153 (1977), 37-50. MR 0447230 (56:5545)
  • 24. F. Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), no. 2, 89-120. MR 597076 (82f:14031)
  • 25. J. P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973. MR 0344216 (49:8956)


Additional Information

Dongseon Hwang
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejon #305-701, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Email: themiso@kaist.ac.kr, dshwang@kias.re.kr

Jonghae Keum
Affiliation: School of Mathematics, Korea Institute For Advanced Study, Seoul 130-722, Korea
Email: jhkeum@kias.re.kr

DOI: https://doi.org/10.1090/S1056-3911-10-00532-1
Received by editor(s): September 12, 2008
Received by editor(s) in revised form: February 9, 2009
Published electronically: March 24, 2010
Additional Notes: Research supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0063180).

American Mathematical Society