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

${\displaystyle p_{g}\leq {\frac {1}{2}}c_{1}(X)^{2}+2.}$

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

${\displaystyle b_{+}\leq 2e+3\sigma +5\,}$

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

${\displaystyle b_{-}+4b_{1}\leq 4b_{+}+9.\,}$

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

${\displaystyle 5c_{1}(X)^{2}-c_{2}(X)+36\geq 12q}$

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

${\displaystyle 5c_{1}(X)^{2}-c_{2}(X)+36\geq 0\quad (c_{1}^{2}(X){\text{ even}})}$

${\displaystyle 5c_{1}(X)^{2}-c_{2}(X)+30\geq 0\quad (c_{1}^{2}(X){\text{ odd}}).}$

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

${\displaystyle 0\to H^{0}({\mathcal {O}}_{X})\to H^{0}(K)\to H^{0}(K|_{D})\to H^{1}({\mathcal {O}}_{X})\to }$

so ${\displaystyle p_{g}-1\leq h^{0}(K|_{D}).}$

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

${\displaystyle h^{0}(K|_{D})-1\leq {\frac {1}{2}}\mathrm {deg} _{D}(K)={\frac {1}{2}}K^{2}.\,}$

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., 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