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In mathematics, the '''Noether inequality''', named after [[Max Noether]], is a property of [[compact space|compact]] minimal [[complex surface]]s that restricts the topological type of the underlying topological [[4-manifold]]. It holds more generally for minimal projective surfaces of general type over an algebraically closed field.


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==Formulation of the inequality==
Let ''X'' be a smooth [[Minimal model (birational geometry)|minimal]] [[Algebraic variety#Projective varieties|projective]] [[surface of general type]] defined over an [[algebraically closed field]] (or a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''<sub>1</sub>(''X''), and let ''p''<sub>g</sub> = ''h''<sup>0</sup>(''K'') be the dimension of the space of holomorphic two forms, then
:<math> p_g \le \frac{1}{2} c_1(X)^2 + 2.  </math>
 
For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented  four manifold. Since a surface of general type is a [[Kaehler manifold|Kähler]] surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by ''b''<sub>+</sub> =&nbsp;1&nbsp;+&nbsp;2''p''<sub>g</sub>. Moreover by the [[Hirzebruch signature theorem]] ''c''<sub>1</sub><sup>2</sup> (''X'')&nbsp;=&nbsp;2''e''&nbsp;+&nbsp;3''σ'', where ''e'' = ''c''<sub>2</sub>(''X'') is the topological [[Euler characteristic]] and ''σ'' = ''b''<sub>+</sub>&nbsp;−&nbsp;''b''<sub></sub> is the signature of the [[intersection form]]. Therefore the Noether inequality can also be expressed as
:<math> b_+ \le 2 e + 3 \sigma + 5 \, </math>
 
or equivalently using ''e'' = 2 – 2 ''b''<sub>1</sub> + ''b''<sub>+</sub> + ''b''<sub>-</sub>
:<math> b_- + 4 b_1 \le 4b_+  + 9. \, </math>
 
Combining the Noether inequality with the [[Noether formula]] 12χ=''c''<sub>1</sub><sup>2</sup>+''c''<sub>2</sub> gives
:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 12q </math>
where ''q'' is the [[irregularity of a surface]], which leads to
a slightly weaker inequality, which is also often called the Noether inequality:
:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text{ even}) </math>
:<math> 5 c_1(X)^2 - c_2(X) + 30 \ge 0 \quad (c_1^2(X)\text{ odd}).  </math>
 
Surfaces where equality holds (i.e. on the Noether line) are called [[Surfaces of general type#Horikawa surfaces|Horikawa surfaces]].
 
==Proof sketch==
It follows from the minimal general type condition that ''K''<sup>2</sup> > 0. We may thus assume that ''p''<sub>g</sub> > 1, since the inequality is otherwise automatic. In particular, we may assume there is an effective divisor ''D'' representing ''K''. We then have an exact sequence
:<math> 0 \to H^0(\mathcal{O}_X) \to H^0(K) \to H^0( K|_D) \to H^1(\mathcal{O}_X) \to   </math>
 
so <math> p_g - 1 \le h^0(K|_D).  </math>
 
Assume that ''D'' is smooth. By the [[adjunction formula]] ''D'' has a canonical linebundle <math>\mathcal{O}_D(2K)</math>, therefore <math>K|_D</math> is a [[special divisor]] and the [[Clifford's theorem on special divisors|Clifford inequality]] applies, which gives
:<math> h^0(K|_D) - 1 \le \frac{1}{2}\mathrm{deg}_D(K) = \frac{1}{2}K^2.\, </math>
 
In general, essentially the same argument applies using a more general version of the Clifford inequality for local complete intersections with a dualising line bundle and 1 dimensional sections in the trivial line bundle.  These conditions are satisfied for the curve ''D'' by the adjunction formula and the fact that ''D'' is numerically connected.
 
==References==
* {{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | id={{MathSciNet | id = 2030225}} | year=2004 | volume=4}}
*{{Citation| last1=Liedtke |first1=Christian|title=Algebraic Surfaces of general type with small c<sub>1</sub><sup>2</sup> in positive characteristic|journal=Nagoya Math. J.|volume=191|year=2008|pages=111–134|
url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1221656783 }}
*{{Citation |doi=10.1007/BF02106598 |last1=Noether |first1 = Max| title=Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde|journal=Math. Ann.|volume=8 |issue=4| year=1875|pages=495–533}}
 
[[Category:Inequalities]]
[[Category:Algebraic surfaces]]

Revision as of 08:40, 20 December 2013

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 of general type over an algebraically closed field.

Formulation of the inequality

Let X be a smooth minimal projective surface of general type defined over an algebraically closed field (or a smooth minimal compact complex surface of general type) with canonical divisor K = −c1(X), and let pg = h0(K) be the dimension of the space of holomorphic two forms, then

For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold. Since a surface of general type is a Kähler surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by b+ = 1 + 2pg. Moreover by the Hirzebruch signature theorem c12 (X) = 2e + 3σ, where e = c2(X) is the topological Euler characteristic and σ = b+ − b is the signature of the intersection form. Therefore the Noether inequality can also be expressed as

or equivalently using e = 2 – 2 b1 + b+ + b-

Combining the Noether inequality with the Noether formula 12χ=c12+c2 gives

where q is the irregularity of a surface, which leads to a slightly weaker inequality, which is also often called the Noether inequality:

Surfaces where equality holds (i.e. on the Noether line) are called Horikawa surfaces.

Proof sketch

It follows from the minimal general type condition that K2 > 0. We may thus assume that pg > 1, since the inequality is otherwise automatic. In particular, we may assume there is an effective divisor D representing K. We then have an exact sequence

so

Assume that D is smooth. By the adjunction formula D has a canonical linebundle , therefore is a special divisor and the Clifford inequality applies, which gives

In general, essentially the same argument applies using a more general version of the Clifford inequality for local complete intersections with a dualising line bundle and 1 dimensional sections in the trivial line bundle. These conditions are satisfied for the curve D by the adjunction formula and the fact that D is numerically connected.

References

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