Bogomolov–Miyaoka–Yau inequality

In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality

[math]\displaystyle{ c_1^2 \le 3 c_2 }[/math]

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

[math]\displaystyle{ c_1^2 \le 3 c_2. }[/math]

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 [math]\displaystyle{ c_2(X) = e(X) }[/math] is the topological Euler characteristic and by the Thom–Hirzebruch signature theorem [math]\displaystyle{ c_1^2(X) = 2 e(X) + 3\sigma(X) }[/math] where [math]\displaystyle{ \sigma(X) }[/math] 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:

[math]\displaystyle{ \sigma(X) \le \frac{1}{3} e(X), }[/math]

moreover if [math]\displaystyle{ \sigma(X) = (1/3)e(X) }[/math] 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 [math]\displaystyle{ c_1^2 = 3 c_2 }[/math], 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 [math]\displaystyle{ {\mathbb C}^2 }[/math] 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), (Prasad Yeung), Donald I. Cartwright and Tim Steger (2010) showed that there are exactly 50 fake projective planes.

(Barthel Hirzebruch) 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., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3 
• 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 
• 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 
• 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 
• Cartwright, Donald I.; Steger, Tim (2010), "Enumeration of the 50 fake projective planes", Comptes Rendus Mathématique (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 
• 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 
• 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 
• 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, Bibcode: 1977InMat..42..225M 
• Mumford, David (1979), "An algebraic surface with K ample, (K²)=9, p_g=q=0", American Journal of Mathematics (The Johns Hopkins University Press) 101 (1): 233–244, doi:10.2307/2373947, ISSN 0002-9327, http://nrs.harvard.edu/urn-3:HUL.InstRepos:3612860 
• Prasad, Gopal; Yeung, Sai-Kee (2007), "Fake projective planes", Inventiones Mathematicae 168 (2): 321–370, doi:10.1007/s00222-007-0034-5, Bibcode: 2007InMat.168..321P 
• Prasad, Gopal; Yeung, Sai-Kee (2010), "Addendum to "Fake projective planes"", Inventiones Mathematicae 182 (1): 213–227, doi:10.1007/s00222-010-0259-6, Bibcode: 2010InMat.182..213P 
• 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, PMID 16578639, Bibcode: 1966PNAS...55.1624V 
• 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, PMID 16592394, Bibcode: 1977PNAS...74.1798Y 
• 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