In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality
:
between
Chern number
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau m ...
s of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
complex surfaces
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with eac ...
of
general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by and , after and proved weaker versions with the constant 3 replaced by 8 and 4.
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
and
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: and 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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
, and let ''c''
1 = ''c''
1(''X'') and ''c''
2 = ''c''
2(''X'') be the first and second
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
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
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswa ...
.
Since
is the topological
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
and by the
Thom–Hirzebruch signature theorem where
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:
:
moreover if
then the universal covering is a ball.
Together with the
Noether inequality In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces ...
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 In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in ...
. see
surfaces of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira ...
.
Surfaces with ''c''12 = 3''c''2
If ''X'' is a surface of general type with
, so that equality holds in the Bogomolov–Miyaoka–Yau inequality, then proved that ''X'' is isomorphic to a quotient of the unit ball in
by an infinite discrete group. Examples of surfaces satisfying this equality are hard to find. showed that there are infinitely many values of ''c'' = 3''c''
2 for which a surface exists. found a
fake projective plane
In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general ...
with ''c'' = 3''c''
2 = 9, which is the minimum possible value because ''c'' + ''c''
2 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'' = 3''c''
2 = 3
25
4.
found a quotient of this surface with ''c'' = 3''c''
2 = 45, and taking unbranched coverings of this quotient gives examples with ''c'' = 3''c''
2 = 45''k'' for any positive integer ''k''.
found examples with ''c'' = 3''c''
2 = 9''n'' for every positive integer ''n''.
References
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{{DEFAULTSORT:Bogomolov-Miyaoka-Yau inequality
Algebraic surfaces
Complex surfaces
Differential geometry
Inequalities