In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality
2 | |
c | |
1 |
\le3c2
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 and, after and 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: and gave examples of surfaces in characteristic p, such as generalized Raynaud surfaces, for which it fails.
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
2 | |
c | |
1 |
\le3c2.
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
c2(X)=e(X)
2(X) | |
c | |
1 |
=2e(X)+3\sigma(X)
\sigma(X)
\sigma(X)\le
1 | |
3 |
e(X),
moreover if
\sigma(X)=(1/3)e(X)
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.
If X is a surface of general type with
2 | |
c | |
1 |
=3c2
{C}2
gave a method for finding examples, which in particular produced a surface X with c = 3c2 = 3254. found a quotient of this surface with c = 3c2 = 45, and taking unbranched coverings of this quotient gives examples with c = 3c2 = 45k for any positive integer k. found examples with c = 3c2 = 9n for every positive integer n.