In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality
c 1 2 ≤ 3 c 2 {\displaystyle c_{1}^{2}\leq 3c_{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 Shing-Tung Yau (1977, 1978) and Yoichi Miyaoka (1977), after Antonius Van de Ven (1966) and Fedor Bogomolov (1978) proved weaker versions with the constant 3 replaced by 8 and 4. Armand Borel and Friedrich Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: William E. Lang (1983) and Robert W. 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 as follows. 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 ≤ 3 c 2 . {\displaystyle c_{1}^{2}\leq 3c_{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 ) {\displaystyle 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 σ ( X ) {\displaystyle c_{1}^{2}(X)=2e(X)+3\sigma (X)} where σ ( X ) {\displaystyle \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:
σ ( X ) ≤ 1 3 e ( X ) , {\displaystyle \sigma (X)\leq {\frac {1}{3}}e(X),}
moreover if σ ( X ) = ( 1 / 3 ) e ( X ) {\displaystyle \sigma (X)=(1/3)e(X)} then the universal covering is a ball. Together 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 c12 = 3c2 If X is a surface of general type with c 1 2 = 3 c 2 {\displaystyle c_{1}^{2}=3c_{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 C 2 {\displaystyle {\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 c21 = 3c2 for which a surface exists. David Mumford (1979) found a fake projective plane with c21 = 3c2 = 9, which is the minimum possible value because c21 + 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 c21 = 3c2 = 3254. Ishida (1988) found a quotient of this surface with c21 = 3c2 = 45, and taking unbranched coverings of this quotient gives examples with c21 = 3c2 = 45k for any positive integer k. Donald I. Cartwright and Tim Steger (2010) found examples with c21 = 3c2 = 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., vol. 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 0912097 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 0522939 Borel, Armand (1963), "Compact Clifford-Klein forms of symmetric spaces", Topology, 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 Mathématique, 348 (1), Elsevier Masson SAS: 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, arXiv:math/0511455, doi:10.1090/S0002-9939-08-09466-5, ISSN 0002-9939, MR 2390492, S2CID 35276117 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 0957050 Lang, William E. (1983), "Examples of surfaces of general type with vector fields", Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Boston, MA: Birkhäuser Boston, pp. 167–173, MR 0717611 Miyaoka, Yoichi (1977), "On the Chern numbers of surfaces of general type", Inventiones Mathematicae, 42 (1): 225–237, Bibcode:1977InMat..42..225M, doi:10.1007/BF01389789, ISSN 0020-9910, MR 0460343, S2CID 120699065 Mumford, David (1979), "An algebraic surface with K ample, (K2)=9, pg=q=0", American Journal of Mathematics, 101 (1), The Johns Hopkins University Press: 233–244, doi:10.2307/2373947, ISSN 0002-9327, JSTOR 2373947, MR 0527834 Prasad, Gopal; Yeung, Sai-Kee (2007), "Fake projective planes", Inventiones Mathematicae, 168 (2): 321–370, arXiv:math/0512115, Bibcode:2007InMat.168..321P, doi:10.1007/s00222-007-0034-5, MR 2289867, S2CID 1990160 Prasad, Gopal; Yeung, Sai-Kee (2010), "Addendum to "Fake projective planes"", Inventiones Mathematicae, 182 (1): 213–227, arXiv:0906.4932, Bibcode:2010InMat.182..213P, doi:10.1007/s00222-010-0259-6, MR 2672284, S2CID 17216453 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, 55 (6), National Academy of Sciences: 1624–1627, Bibcode:1966PNAS...55.1624V, doi:10.1073/pnas.55.6.1624, ISSN 0027-8424, JSTOR 57245, MR 0198496, PMC 224368, PMID 16578639 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, 74 (5), National Academy of Sciences: 1798–1799, Bibcode:1977PNAS...74.1798Y, doi:10.1073/pnas.74.5.1798, ISSN 0027-8424, JSTOR 67110, MR 0451180, PMC 431004, PMID 16592394 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 0480350