In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality
Contents
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
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
moreover if
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
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 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.