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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]]

Latest revision as of 02:14, 13 October 2014

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