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

$p_g\le \frac{1}{2}{c}_1(X{)}^2+2.$

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

$b_{+}\le 2e+3\sigma +5$

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

$b_{-}+4b_1\le 4b_{+}+9.$

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

$5c_1(X{)}^2-c_2(X)+36\ge 12q$

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

$5c_1(X{)}^2-c_2(X)+36\ge 0(c_1^2(X)\text{ even})$

$5c_1(X{)}^2-c_2(X)+30\ge 0(c_1^2(X)\text{ odd}).$

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

$0\to H^0({{𝒪}}_X)\to H^0(K)\to H^0(K{|}_D)\to H^1({{𝒪}}_X)\to$

so $p_g-1\le h^0(K{|}_D).$ Assume that D is smooth. By the adjunction formula D has a canonical linebundle ${{𝒪}}_D(2K)$, therefore $K{|}_D$ is a special divisor and the Clifford inequality applies, which gives

$h^0(K{|}_D)-1\le \frac{1}{2}{\mathrm{deg}}_D(K)=\frac{1}{2}{K}^2.$

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., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Liedtke, Christian (2008), "Algebraic Surfaces of general type with small c₁² in positive characteristic", Nagoya Math. J., 191: 111–134
• Noether, Max (1875), "Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde", Math. Ann., 8 (4): 495–533, doi:10.1007/BF02106598