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 BogomolovâMiyaokaâYau 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
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 is the topological
Euler characteristic and by the
ThomâHirzebruch signature theorem where is the signature of the
intersection form on the second cohomology, the BogomolovâMiyaokaâYau inequality can also be written as a restriction on the topological type of the surface of general type:
moreover if then the universal covering is a ball.
Together with the
Noether inequality the BogomolovâMiyaokaâYau 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 , so that equality holds in the BogomolovâMiyaokaâYau inequality, then
Yau (1977) proved that X is isomorphic to a quotient of the unit ball in 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.
David 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 c2 1 = 3c2 = 3254.
Ishida (1988) found a quotient of this surface with c2 1 = 3c2 = 45, and taking unbranched coverings of this quotient gives examples with c2 1 = 3c2 = 45k for any positive integer k.
Donald I. Cartwright and Tim Steger (
2010) found examples with c2 1 = 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,
ISBN978-3-540-00832-3,
MR2030225
Barthel, Gottfried;
Hirzebruch, Friedrich; Höfer, Thomas (1987), Geradenkonfigurationen und Algebraische FlÀchen, Aspects of Mathematics, D4, Braunschweig: Friedr. Vieweg & Sohn,
ISBN978-3-528-08907-8,
MR0912097
Bogomolov, Fedor A. (1978), "Holomorphic tensors and vector bundles on projective manifolds", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 42 (6): 1227â1287,
ISSN0373-2436,
MR0522939
Ishida, Masa-Nori (1988), "An elliptic surface covered by Mumford's fake projective plane", The Tohoku Mathematical Journal, Second Series, 40 (3): 367â396,
doi:10.2748/tmj/1178227980,
ISSN0040-8735,
MR0957050
Lang, William E. (1983), "Examples of surfaces of general type with vector fields", Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Boston, MA: BirkhĂ€user Boston, pp. 167â173,
MR0717611