Bogomolov–Miyaoka–Yau 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 : 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. see surfaces of general type. ==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 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. showed that there are infinitely many values of c = 3c2 for which a surface exists. found a fake projective plane with c = 3c2 = 9, which is the minimum possible value because c + c2 is always divisible by 12, and , , showed that there are exactly 50 fake projective planes. 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. ==References==