## A vanishing theorem on fake projective planes with enough automorphisms

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- by JongHae Keum PDF
- Trans. Amer. Math. Soc.
**369**(2017), 7067-7083 Request permission

## 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

- 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**433520** - 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**, DOI 10.1016/j.crma.2009.11.016 - D. Cartwright and T. Steger, http://www.maths.usyd.edu.au/u/donaldc/fakeprojective planes
- 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**, DOI 10.1007/978-3-642-11087-0_{3} - Najmuddin Fakhruddin,
*Exceptional collections on 2-adically uniformized fake projective planes*, Math. Res. Lett.**22**(2015), no. 1, 43–57. MR**3342178**, DOI 10.4310/MRL.2015.v22.n1.a4 - 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**, DOI 10.1090/S1056-3911-10-00532-1 - DongSeon Hwang and JongHae Keum,
*Algebraic Montgomery-Yang problem: the nonrational surface case*, Michigan Math. J.**62**(2013), no. 1, 3–37. MR**3049295**, DOI 10.1307/mmj/1363958239 - S. Galkin, L. Katzarkov, A. Mellit, and E. Shinder,
*Minifolds and Phantoms*, arXiv:1305.4549. - 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**, DOI 10.2748/tmj/1178227980 - JongHae Keum,
*A fake projective plane with an order 7 automorphism*, Topology**45**(2006), no. 5, 919–927. MR**2239523**, DOI 10.1016/j.top.2006.06.006 - Jonghae Keum,
*Quotients of fake projective planes*, Geom. Topol.**12**(2008), no. 4, 2497–2515. MR**2443971**, DOI 10.2140/gt.2008.12.2497 - 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**, DOI 10.1007/s11425-011-4247-0 - Jonghae Keum,
*Toward a geometric construction of fake projective planes*, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.**23**(2012), no. 2, 137–155. MR**2924897**, DOI 10.4171/RLM/622 - 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**, DOI 10.1007/s00222-002-0283-2 - János Kollár,
*Shafarevich maps and automorphic forms*, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR**1341589**, DOI 10.1515/9781400864195 - 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**, DOI 10.2307/2373947 - Gopal Prasad and Sai-Kee Yeung,
*Fake projective planes*, Invent. Math.**168**(2007), no. 2, 321–370. MR**2289867**, DOI 10.1007/s00222-007-0034-5 - 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**451180**, DOI 10.1073/pnas.74.5.1798

## Additional Information

**JongHae Keum**- Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Dondaemungu, Seoul 02455, Korea
- MR Author ID: 291447
- Email: jhkeum@kias.re.kr
- 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)
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 7067-7083 - MSC (2010): Primary 14J29, 14F05
- DOI: https://doi.org/10.1090/tran/6856
- MathSciNet review: 3683103