Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A vanishing theorem on fake projective planes with enough automorphisms


Author: JongHae Keum
Journal: Trans. Amer. Math. Soc. 369 (2017), 7067-7083
MSC (2010): Primary 14J29, 14F05
DOI: https://doi.org/10.1090/tran/6856
Published electronically: March 29, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For every fake projective plane $ X$ with automorphism group of order 21, we prove that $ H^i(X, 2L)=0$ for all $ i$ and for every ample line bundle $ L$ with $ L^2=1$. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle $ K$. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes.


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

  • [Au] Thierry Aubin, Équations du type Monge-Ampère sur les variétés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Aiii, A119-A121. MR 0433520
  • [CS] 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, https://doi.org/10.1016/j.crma.2009.11.016
  • [CS2] D. Cartwright and T. Steger, http://www.maths.usyd.edu.au/u/donaldc/fakeprojective planes
  • [D] Igor Dolgachev, Algebraic surfaces with $ q=p_g=0$, Algebraic surfaces, C.I.M.E. Summer Sch., vol. 76, Springer, Heidelberg, 2010, pp. 97-215. MR 2757651, https://doi.org/10.1007/978-3-642-11087-0_3
  • [F] Najmuddin Fakhruddin, Exceptional collections on 2-adically uniformized fake projective planes, Math. Res. Lett. 22 (2015), no. 1, 43-57. MR 3342178, https://doi.org/10.4310/MRL.2015.v22.n1.a4
  • [HK1] Dongseon Hwang and Jonghae Keum, The maximum number of singular points on rational homology projective planes, J. Algebraic Geom. 20 (2011), no. 3, 495-523. MR 2786664, https://doi.org/10.1090/S1056-3911-10-00532-1
  • [HK2] DongSeon Hwang and JongHae Keum, Algebraic Montgomery-Yang problem: the nonrational surface case, Michigan Math. J. 62 (2013), no. 1, 3-37. MR 3049295, https://doi.org/10.1307/mmj/1363958239
  • [GKMS] S. Galkin, L. Katzarkov, A. Mellit, and E. Shinder, Minifolds and Phantoms, arXiv:1305.4549.
  • [I] Masa-Nori Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math. J. (2) 40 (1988), no. 3, 367-396. MR 957050, https://doi.org/10.2748/tmj/1178227980
  • [K06] JongHae Keum, A fake projective plane with an order 7 automorphism, Topology 45 (2006), no. 5, 919-927. MR 2239523, https://doi.org/10.1016/j.top.2006.06.006
  • [K08] Jonghae Keum, Quotients of fake projective planes, Geom. Topol. 12 (2008), no. 4, 2497-2515. MR 2443971, https://doi.org/10.2140/gt.2008.12.2497
  • [K11] JongHae Keum, A fake projective plane constructed from an elliptic surface with multiplicities $ (2,4)$, Sci. China Math. 54 (2011), no. 8, 1665-1678. MR 2824965, https://doi.org/10.1007/s11425-011-4247-0
  • [K12] Jonghae Keum, Toward a geometric construction of fake projective planes, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 23 (2012), no. 2, 137-155. MR 2924897, https://doi.org/10.4171/RLM/622
  • [Kl] Bruno Klingler, Sur la rigidité de certains groupes fondamentaux, l'arithméticité des réseaux hyperboliques complexes, et les ``faux plans projectifs'', Invent. Math. 153 (2003), no. 1, 105-143 (French, with English summary). MR 1990668, https://doi.org/10.1007/s00222-002-0283-2
  • [Ko] János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589
  • [M] D. Mumford, An algebraic surface with $ K$ ample, $ (K^{2})=9$, $ p_{g}=q=0$, Amer. J. Math. 101 (1979), no. 1, 233-244. MR 527834, https://doi.org/10.2307/2373947
  • [PY] Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321-370. MR 2289867, https://doi.org/10.1007/s00222-007-0034-5
  • [Y] Shing Tung Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798-1799. MR 0451180

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14J29, 14F05

Retrieve articles in all journals with MSC (2010): 14J29, 14F05


Additional Information

JongHae Keum
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dondaemungu, Seoul 02455, Korea
Email: jhkeum@kias.re.kr

DOI: https://doi.org/10.1090/tran/6856
Received by editor(s): February 5, 2015
Received by editor(s) in revised form: October 20, 2015
Published electronically: March 29, 2017
Additional Notes: This research was supported by the National Research Foundation of Korea (NRF-2007-0093858)
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society