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.
 [B]
F.
A. Bogomolov, Holomorphic tensors and vector bundles on projective
manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978),
no. 6, 1227–1287, 1439 (Russian). MR 522939
(80j:14014)
 [C]
F.
Catanese, On the moduli spaces of surfaces of general type, J.
Differential Geom. 19 (1984), no. 2, 483–515.
MR 755236
(86h:14031)
 [H1]
Friedrich
Hirzebruch, Automorphe Formen und der Satz von RiemannRoch,
Symposium internacional de topología algebraica International
symposium on algebraic topology, Universidad Nacional Autónoma de
México and UNESCO, Mexico City, 1958, pp. 129–144
(German). MR
0103280 (21 #2058)
 [H2]
F.
Hirzebruch, Arrangements of lines and algebraic surfaces,
Arithmetic and geometry, Vol. II, Progr. Math., vol. 36,
Birkhäuser, Boston, Mass., 1983, pp. 113–140. MR 717609
(84m:14037)
 [H3]
F.
Hirzebruch, Chern numbers of algebraic surfaces: an example,
Math. Ann. 266 (1984), no. 3, 351–356. MR 730175
(85j:14069), http://dx.doi.org/10.1007/BF01475584
 [L]
William
E. Lang, Examples of surfaces of general type with vector
fields, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36,
Birkhäuser Boston, Boston, MA, 1983, pp. 167–173. MR 717611
(86f:14022)
 [M1]
Marco
Manetti, On the moduli space of diffeomorphic algebraic
surfaces, Invent. Math. 143 (2001), no. 1,
29–76. MR
1802792 (2001j:14050), http://dx.doi.org/10.1007/s002220000101
 [M2]
Yoichi
Miyaoka, On the Chern numbers of surfaces of general type,
Invent. Math. 42 (1977), 225–237. MR 0460343
(57 #337)
 [P]
Rita
Pardini, Abelian covers of algebraic varieties, J. Reine
Angew. Math. 417 (1991), 191–213. MR 1103912
(92g:14012), http://dx.doi.org/10.1515/crll.1991.417.191
 [R]
Miles
Reid, Bogomolov’s theorem
𝑐₁²≤4𝑐₂, Proceedings of the
International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977)
Kinokuniya Book Store, Tokyo, 1978, pp. 623–642. MR 578877
(82b:14014)
 [V]
R. Vakil, Murphy's law in algebraic geometry: Badlybehaved deformation spaces, Invent. Math., to appear.
 [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
(56 #9467)
 [B]
 F.A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Math. USSRIzv. 13 (1979), 499555. MR 0522939 (80j:14014)
 [C]
 F. Catanese, On the moduli spaces of surfaces of general type, J. Diff. Geom. 19 (1984), 483515. MR 755236 (86h:14031)
 [H1]
 F. Hirzebruch, Automorphe Formen und der Satz von RiemannRoch, in Symposium Internacional de Topologia Algebraica, Mexico (1958). MR 0103280 (21:2058)
 [H2]
 , Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, Mass. (1983), 113140. MR 717609 (84m:14037)
 [H3]
 , Chern numbers of algebraic surfaces: an example, Math. Ann. 266, no. 3 (1984), 351356. MR 730175 (85j:14069)
 [L]
 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)
 [M1]
 M. Manetti, On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143 (2001), 2976. MR 1802792 (2001j:14050)
 [M2]
 Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225237. MR 0460343 (57:337)
 [P]
 R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191213. MR 1103912 (92g:14012)
 [R]
 M. Reid, Bogomolov's theorem , in Proceedings of the International Symposium on Algebraic Geometry, Kinokuniya Book Store, Tokyo (1978), 623642. MR 578877 (82b:14014)
 [V]
 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.
