Bogomolov–Miyaoka–Yau inequality

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

$c_1^2 \le 3 c_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 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 c1c1(X) and c2c2(X) be the first and second Chern class of the complex tangent bundle of the surface. Then

$c_1^2 \le 3 c_2. \,$

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

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

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

Surfaces with c12 = 3c2

If X is a surface of general type with $c_1^2 = 3 c_2$, 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 ${\mathbb C}^2$ by an infinite discrete group. Examples of surfaces satisfying this equality are hard to find. Borel (1963) showed that there are infinitely many values of c2
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 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.