Bogomolov Inequality - Miaoki - Yau

Bogomolov-Miaoki-Yau inequality is inequality

c_{1}^{2}\leqslant 3c_{2}

{\ displaystyle c_ {1} ^ {2} \ leqslant 3c_ {2}}

{\ displaystyle c_ {1} ^ {2} \ leqslant 3c_ {2}}

between compact complex surfaces of general form . The main interest in this inequality is the possibility to limit the possible topological types of the real 4-manifold under consideration. Inequality was proved independently by Yau ^[1] ^[2] and Miaoki ^[3] , after Van de Ven ^[4] and Fyodor Bogomol ^[5] proved weaker versions of inequality with constants 8 and 4 instead of 3.

Borel and Hirzebruch showed that inequality cannot be improved by finding infinitely many cases in which equality holds. The inequality is not true for positive characteristics — Leng ^[6] and Easton ^[7] gave examples of surfaces with characteristic p , such as the , for which the inequality is not satisfied.

Content

Inequality

Usually the Bogomolov – Miaoki – Yau inequality is formulated as follows.

Let X be a compact complex surface of a , and let $c_{1}=c_{1}(X)$ ${\ displaystyle c_ {1} = c_ {1} (X)}$ and $c_{2}=c_{2}(X)$ ${\ displaystyle c_ {2} = c_ {2} (X)}$ - the first and second complex tangent surface stratification. Then

c_{1}^{2}\leqslant 3c_{2}.

{\ displaystyle c_ {1} ^ {2} \ leqslant 3c_ {2}.}

Moreover, if equality holds, then X is a factor of the ball. The last statement is a consequence of Yau's approach in differential geometry, which is based on his resolution .

Insofar as $c_{2}(X)=e(X)$ ${\ displaystyle c_ {2} (X) = e (X)}$ is a topological characteristic of Euler , and by $c_{1}^{2}(X)=2e(X)+3\sigma (X)$ ${\ displaystyle c_ {1} ^ {2} (X) = 2e (X) +3 \ sigma (X)}$ where $\sigma (X)$ ${\ displaystyle \ sigma (X)}$ is the signature of the intersection form on the second cohomology, the Bogomolov – Miaoki – Yau inequality can be rewritten as a restriction on the topological type of a general surface:

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

{\ displaystyle \ sigma (X) \ leqslant {\ frac {1} {3}} e (X),}

and moreover if $\sigma (X)=(1/3)e(X)$ ${\ displaystyle \ sigma (X) = (1/3) e (X)}$ The universal cover is a ball.

Together with inequality, the Bogomolov – Miaoki – Yau inequality establishes boundaries when searching for complex surfaces. Consideration of topological types that can be implemented as complex surfaces is called . See the article .

Surfaces with c ₁ ² = 3 c ₂

Let X be a surface of general type with $c_{1}^{2}=3c_{2}$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2}}$ , so that in the inequality of Bogomolov - Miaoki - Yau equality takes place. For such surfaces, Yau ^[1] proved that X is isomorphic to the unit ball factor in ${\mathbb {C} }^{2}$ ${\ displaystyle {\ mathbb {C}} ^ {2}}$ by infinite discrete group. Examples of surfaces for which equality holds are difficult to find. Borel ^[8] showed that there are infinitely many values $c_{1}^{2}=3c_{2}$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2}}$ for which surfaces exist. Mumford ^[9] found a false projective plane with $c_{1}^{2}=3c_{2}=9$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2} = 9}$ which has the lowest possible value because $c_{1}^{2}+c_{2}$ ${\ displaystyle c_ {1} ^ {2} + c_ {2}}$ always divisible by 12, while Prasad and Yen ^[10] ^[11] , as well as Cartwright and Steger ^[12] showed that there are exactly 50 false projective surfaces.

Bartel, Hirzebruch, and Höfer ^[13] gave a method of searching for examples, which, in particular, gives surfaces X with $c_{1}^{2}=3c_{2}=3^{2}5^{4}$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2} = 3 ^ {2} 5 ^ {4}}$ . Ishida ^[14] found the factor of such a surface with $c_{1}^{2}=3c_{2}=45$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2} = 45}$ and if we take the unbranched coverage of this factor, we get examples with $c_{1}^{2}=3c_{2}=45$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2} = 45}$ for any positive k . Cartwright and Steger ^[12] found examples with $c_{1}^{2}=3c_{2}=9n$ ${\ displaystyle c_ {1} ^ {2} = 3c_ {2} = 9n}$ for any positive integer n .

Notes

↑ ¹ ² Yau, 1977 .
↑ Yau, 1978 .
↑ Miyaoka, 1977 .
↑ Van de Ven, 1966 .
↑ Bogomolov, 1978 .
↑ Lang, 1983 .
↑ Easton, 2008 .
↑ Borel, 1963 .
↑ Mumford, 1979 .
↑ Prasad, Yeung, 2007 .
↑ Prasad, Yeung, 2010 .
↑ ¹ ² Cartwright, Steger, 2010 , p. 11–13.
↑ Barthel, Hirzebruch, Höfer, 1987 .
↑ Ishida, 1988 .

Literature

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Donald I. Cartwright, Tim Steger. Enumeration of the 50 fake projective planes // Comptes Rendus Mathematique. - Elsevier Masson SAS, 2010. - V. 348 , no. 1 . - pp . 11–13 . - DOI : 10.1016 / j.crma.2009.11.016 .
Wolf P. Barth, Klaus Hulek, Chris AM Peters, Antonius Van de Ven. Compact Complex Surfaces. - Springer-Verlag, Berlin, 2004. - Vol. 4. - (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.). - ISBN 978-3-540-00832-3 .
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[_354198bcb50fe132-1] ¹ ² Yau, 1977 .

[_bc405d20745190e9-2] Yau, 1978 .

[_f12120360af9f5de-3] Miyaoka, 1977 .

[_289a170434b1959e-4] Van de Ven, 1966 .

[_6f5231fb655b775e-5] Bogomolov, 1978 .

[_beb3dda5af886f90-6] Lang, 1983 .

[_abd74cd7ac864a47-7] Easton, 2008 .

[_90d0c6d1c761223e-8] Borel, 1963 .

[_044e5ab3087a5365-9] Mumford, 1979 .

[_dc92acdc2c95dd79-10] Prasad, Yeung, 2007 .

[_1b4b7fd2e2e24ef1-11] Prasad, Yeung, 2010 .

[_31ef33752f70d752-12] ¹ ² Cartwright, Steger, 2010 , p. 11–13.

[_8f25d2b7225341ee-13] Barthel, Hirzebruch, Höfer, 1987 .

[_49eb8792941608bf-14] Ishida, 1988 .