Noether inequality

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 \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 + 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_+ \le 2 e + 3 \sigma + 5 \,

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

 b_- + 4 b_1 \le 4b_+  + 9. \,

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

  5 c_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:

  5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text{ even})
  5 c_1(X)^2 - c_2(X) + 30 \ge 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 \le 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 \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


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