Bogomolov–Miyaoka–Yau inequality

${\displaystyle c_1^2\le 3c_2}$

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

${\displaystyle c_1^2\le 3c_2.}$

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

${\displaystyle \sigma (X)\le \frac{1}{3}e(X),}$

moreover if ${\displaystyle \sigma (X)=(1/3)e(X)}$ 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.