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Bogomolov–Miyaoka–Yau inequality
In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality
\( c_1^2 \le 3 c_2\ \)
between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by S.-T. Yau (1977, 1978) and Yoichi Miyaoka (1977), after Van de Ven (1966) and Fedor Bogomolov (1978) proved weaker versions with the constant 3 replaced by 8 and 4.
Borel and Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: (Lang 1983) and Easton (2008) gave examples of surfaces in characteristic p, such as generalized Raynaud surfaces, for which it fails.
Formulation of the inequality
The conventional formulation of the Bogomolov–Miyaoka–Yau inequality is
Let X be a compact complex surface of general type, and let c1 = c1(X) and c2 = c2(X) be the first and second Chern class of the complex tangent bundle of the surface. Then
\( c_1^2 \le 3 c_2. \, \)
moreover if equality holds then X is a quotient of a ball. The latter statement is a consequence of Yau's differential geometric approach which is based on his resolution of the Calabi conjecture.
Since \( c_2(X) = e(X) is the topological Euler characteristic and by the Thom–Hirzebruch signature theorem \(c_1^2(X) = 2 e(X) + 3\sigma(X) where \sigma(X) is the signature of the intersection form on the second cohomology, the Bogomolov–Miyaoka–Yau inequality can also be written as a restriction on the topological type of the surface of general type:
\( \sigma(X) \le \frac{1}{3} e(X),
moreover if \( \sigma(X) = (1/3)e(X) \) then the universal covering is a ball.
ogether with the Noether inequality the Bogomolov–Miyaoka–Yau inequality sets boundaries in the search for complex surfaces. Mapping out the topological types that are realized as complex surfaces is called geography of surfaces. see surfaces of general type.
Surfaces with \(c_1^2 = 3 c_2 \)
If X is a surface of general type with \(c_1^2 = 3 c_2 \), so that equality holds in the Bogomolov–Miyaoka–Yau inequality, then Yau (1977) proved that X is isomorphic to a quotient of the unit ball in \( {\mathbb C}^2 \) by an infinite discrete group.
Examples of surfaces satisfying this equality are hard to find. Borel (1963) showed that there are infinitely many values of c2
1 = 3c2 for which a surface exists. Mumford (1979) found a fake projective plane with c2
1 = 3c2 = 9, which is the minimum possible value because c2
1 + c2 is always divisible by 12, and Prasad & Yeung (2007), Prasad & Yeung (2010), Donald I. Cartwright and Tim Steger (2010) showed that there are exactly 50 fake projective planes.
Barthel, Hirzebruch & Höfer (1987) gave a method for finding examples, which in particular produced a surface X with \( c_1^2 = 3c_2 = 3^25^4\). Ishida (1988) found a quotient of this surface with \( c_1^2 = 3c_2 = 45 \), and taking unbranched coverings of this quotient gives examples with \( c_1^2 = 3c_2 = 45k \) for any positive integer k. Donald I. Cartwright and Tim Steger (2010) found examples with \( c_1^2 = 3c_2 = 9n \) for every positive integer n.
References
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
Barthel, Gottfried; Hirzebruch, Friedrich; Höfer, Thomas (1987), Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Braunschweig: Friedr. Vieweg & Sohn, ISBN 978-3-528-08907-8, MR 912097
Bogomolov, Fedor A. (1978), "Holomorphic tensors and vector bundles on projective manifolds", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 42 (6): 1227–1287, ISSN 0373-2436, MR 522939
Borel, Armand (1963), "Compact Clifford-Klein forms of symmetric spaces", Topology. an International Journal of Mathematics 2 (1-2): 111–122, doi:10.1016/0040-9383(63)90026-0, ISSN 0040-9383, MR 0146301
Cartwright, Donald I.; Steger, Tim (2010), "Enumeration of the 50 fake projective planes", Comptes Rendus Mathematique (Elsevier Masson SAS) 348 (1): 11–13, doi:10.1016/j.crma.2009.11.016
Easton, Robert W. (2008), "Surfaces violating Bogomolov-Miyaoka-Yau in positive characteristic", Proceedings of the American Mathematical Society 136 (7): 2271–2278, doi:10.1090/S0002-9939-08-09466-5, ISSN 0002-9939, MR 2390492
Ishida, Masa-Nori (1988), "An elliptic surface covered by Mumford's fake projective plane", The Tohoku Mathematical Journal. Second Series 40 (3): 367–396, doi:10.2748/tmj/1178227980, ISSN 0040-8735, MR 957050
Lang, William E. (1983), "Examples of surfaces of general type with vector fields", Arithmetic and geometry, Vol. II, Progr. Math. 36, Boston, MA: Birkhäuser Boston, pp. 167–173, MR 717611
Miyaoka, Yoichi (1977), "On the Chern numbers of surfaces of general type", Inventiones Mathematicae 42 (1): 225–237, doi:10.1007/BF01389789, ISSN 0020-9910, MR 0460343
Mumford, David (1979), "An algebraic surface with K ample, (K2)=9, pg=q=0", American Journal of Mathematics (The Johns Hopkins University Press) 101 (1): 233–244, doi:10.2307/2373947, ISSN 0002-9327, JSTOR 2373947, MR 527834
Prasad, Gopal; Yeung, Sai-Kee (2007), "Fake projective planes", Inventiones Mathematicae 168 (2): 321–370, arXiv:math/0512115, doi:10.1007/s00222-007-0034-5, MR 2289867
Prasad, Gopal; Yeung, Sai-Kee (2010), "Addendum to "Fake projective planes"", Inventiones Mathematicae 182 (1): 213–227, doi:10.1007/s00222-010-0259-6, MR 2672284
Van de Ven, Antonius (1966), "On the Chern numbers of certain complex and almost complex manifolds", Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences) 55 (6): 1624–1627, doi:10.1073/pnas.55.6.1624, ISSN 0027-8424, JSTOR 57245, MR 0198496
Yau, Shing Tung (1977), "Calabi's conjecture and some new results in algebraic geometry", Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences) 74 (5): 1798–1799, doi:10.1073/pnas.74.5.1798, ISSN 0027-8424, JSTOR 67110, MR 0451180
Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I", Communications on Pure and Applied Mathematics 31 (3): 339–411, doi:10.1002/cpa.3160310304, ISSN 0010-3640, MR 480350
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