Bogomolov–Miyaoka–Yau inequality

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 {{harvs|txt|last=Yau|first=Shing-Tung|authorlink=Shing-Tung Yau| year1=1977|year2=1978}} and {{harvs|txt|authorlink=Yoichi Miyaoka|first=Yoichi|last=Miyaoka|year=1977}}, after {{harvs|txt|last=Van de Ven|first=Antonius|year=1966}} and {{harvs|txt|authorlink=Fedor Bogomolov|first=Fedor|last=Bogomolov|year=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: {{harvs|txt|first=William E. |last=Lang|year=1983}} and {{harvs|txt|last=Easton|first=Robert W.|year=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

: $c_1^2 \le 3 c_2.$

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 ''c''12 = 3''c''2

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 {{harvtxt|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. {{harvtxt|Borel|1963}} showed that there are infinitely many values of c{{sup sub|2|1}} = 3c₂ for which a surface exists. {{harvs|txt|last=Mumford|first=David|authorlink=David Mumford|year=1979}} found a fake projective plane with c{{sup sub|2|1}} = 3c₂ = 9, which is the minimum possible value because c{{sup sub|2|1}} + c₂ is always divisible by 12, and {{harvtxt|Prasad|Yeung|2007}}, {{harvtxt|Prasad|Yeung|2010}}, {{harvs|txt | last1=Cartwright | first1=Donald I. | last2=Steger | first2=Tim | title=Enumeration of the 50 fake projective planes | publisher=Elsevier Masson SAS | doi=10.1016/j.crma.2009.11.016 | journal=Comptes Rendus Mathématique | volume=348 | issue=1 | pages=11–13|year=2010}} showed that there are exactly 50 fake projective planes.

{{harvtxt|Barthel|Hirzebruch|Höfer|1987}} gave a method for finding examples, which in particular produced a surface X with c{{sup sub|2|1}} = 3c₂ = 3²5⁴.

{{harvtxt|Ishida|1988}} found a quotient of this surface with c{{sup sub|2|1}} = 3c₂ = 45, and taking unbranched coverings of this quotient gives examples with c{{sup sub|2|1}} = 3c₂ = 45k for any positive integer k.

{{harvs|txt | last1=Cartwright | first1=Donald I. | last2=Steger | first2=Tim | title=Enumeration of the 50 fake projective planes | publisher=Elsevier Masson SAS | doi=10.1016/j.crma.2009.11.016 | journal=Comptes Rendus Mathématique | volume=348 | issue=1 | pages=11–13|year=2010}} found examples with c{{sup sub|2|1}} = 3c₂ = 9n for every positive integer n.

References

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Category:Algebraic surfaces

Category:Complex surfaces

Category:Differential geometry

Category:Inequalities (mathematics)

Bogomolov–Miyaoka–Yau inequality

Formulation of the inequality

Surfaces with ''c''<sub>1</sub><sup>2</sup> = 3''c''<sub>2</sub>

References