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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
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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.


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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