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.
The latter statement is a consequence of Yau's differential geometric approach which is based on his resolution of the Calabi conjecture.
, 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
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.