Surfaces violating BogomolovMiyaokaYau in positive characteristic
Author:
Robert W. Easton
Journal:
Proc. Amer. Math. Soc. 136 (2008), 22712278
MSC (2000):
Primary 14J29
Published electronically:
March 6, 2008
MathSciNet review:
2390492
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Abstract: The BogomolovMiyaokaYau inequality asserts that the Chern numbers of a surface of general type in characteristic 0 satisfy the inequality , a consequence of which is . This inequality fails in characteristic , and here we produce infinite families of counterexamples for large . Our method parallels a construction of Hirzebruch, and relies on a construction of abelian covers due to Catanese and Pardini.
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Catanese, On the moduli spaces of surfaces of general type, J.
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 , Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, Mass. (1983), 113140. MR 717609 (84m:14037)
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 , Chern numbers of algebraic surfaces: an example, Math. Ann. 266, no. 3 (1984), 351356. MR 730175 (85j:14069)
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 W.E. Lang, Examples of surfaces of general type with vector fields, in Arithmetic and Geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, Mass. (1983), 167173. MR 717611 (86f:14022)
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 M. Manetti, On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143 (2001), 2976. MR 1802792 (2001j:14050)
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 Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225237. MR 0460343 (57:337)
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 R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191213. MR 1103912 (92g:14012)
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 M. Reid, Bogomolov's theorem , in Proceedings of the International Symposium on Algebraic Geometry, Kinokuniya Book Store, Tokyo (1978), 623642. MR 578877 (82b:14014)
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 R. Vakil, Murphy's law in algebraic geometry: Badlybehaved deformation spaces, Invent. Math., to appear.
 [Y]
 S.T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA 74 (1977), 17981799. MR 0451180 (56:9467)
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Additional Information
Robert W. Easton
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84102
Email:
easton@math.utah.edu
DOI:
http://dx.doi.org/10.1090/S0002993908094665
PII:
S 00029939(08)094665
Keywords:
Bogomolov inequality,
abelian cover,
positive characteristic,
algebraic surface,
general type
Received by editor(s):
December 6, 2005
Published electronically:
March 6, 2008
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
