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 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 p_g = h⁰(K) be the dimension of the space of holomorphic two forms, then

[math]\displaystyle{ 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 Kähler surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by b₊ = 1 + 2p_g. 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

[math]\displaystyle{ b_+ \le 2 e + 3 \sigma + 5 }[/math]

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

[math]\displaystyle{ b_- + 4 b_1 \le 4b_+ + 9. }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ 5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text{ even}) }[/math]

[math]\displaystyle{ 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 Horikawa surfaces.

Proof sketch

It follows from the minimal general type condition that K² > 0. We may thus assume that p_g > 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]\displaystyle{ 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]\displaystyle{ p_g - 1 \le h^0(K|_D). }[/math]

Assume that D is smooth. By the adjunction formula D has a canonical linebundle [math]\displaystyle{ \mathcal{O}_D(2K) }[/math], therefore [math]\displaystyle{ K|_D }[/math] is a special divisor and the Clifford inequality applies, which gives

[math]\displaystyle{ h^0(K|_D) - 1 \le \frac{1}{2} \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

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3 
• Liedtke, Christian (2008), "Algebraic Surfaces of general type with small c₁² in positive characteristic", Nagoya Math. J. 191: 111–134, http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1221656783 
• Noether, Max (1875), "Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde", Math. Ann. 8 (4): 495–533, doi:10.1007/BF02106598