Bogomolov–Miyaoka–Yau inequality

In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality

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.

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 ${\displaystyle c_{2}(X)=e(X)}$ 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:

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