Noether inequality explained

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 (1)/(2) c ^{2} _{1 (X)} + 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 2 e + 3 \sigma + 5

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

b _{-} + 4 b _{1} \le 4 b _{+} + 9 .

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

5 c ^{2} _{1 (X)} - c _{2(X)} + 36 \ge 12 q where q is the irregularity of a surface, which leads toa slightly weaker inequality, which is also often called the Noether inequality:5 c ^{2} _{1 (X)} - c _{2(X)} + 36 \ge 0 (c ^{2 (X) even )} _{1} 5 c ^{2} _{1 (X)} - c _{2(X)} + 30 \ge 0 (c ^{2 (X) odd ).} _{1}

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

l{O} _{D(2K)} , therefore K| _{D} is a special divisor and the Clifford inequality applies, which givesh ^{0 (K|} _{D)} - 1 \le (1)/(2) \deg _{D(K)} = (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.