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

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

$b_{+}\leq 2e+3\sigma +5$ or equivalently using e = 2 – 2 b1 + b+ + b

$b_{-}+4b_{1}\leq 4b_{+}+9.$ Combining the Noether inequality with the Noether formula 12χ=c12+c2 gives

$5c_{1}(X)^{2}-c_{2}(X)+36\geq 12q$ where q is the irregularity of a surface, which leads to a slightly weaker inequality, which is also often called the Noether inequality:

$5c_{1}(X)^{2}-c_{2}(X)+36\geq 0\quad (c_{1}^{2}(X){\text{ even}})$ $5c_{1}(X)^{2}-c_{2}(X)+30\geq 0\quad (c_{1}^{2}(X){\text{ odd}}).$ 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 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

$0\to H^{0}({\mathcal {O}}_{X})\to H^{0}(K)\to H^{0}(K|_{D})\to H^{1}({\mathcal {O}}_{X})\to$ so $p_{g}-1\leq h^{0}(K|_{D}).$ Assume that D is smooth. By the adjunction formula D has a canonical linebundle ${\mathcal {O}}_{D}(2K)$ , therefore $K|_{D}$ is a special divisor and the Clifford inequality applies, which gives

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