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Noether inequality

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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 = h(K) be the dimension of the space of holomorphic two forms, then

p g ≤ 1 2 c 1 ( X ) 2 + 2. {\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 + 2pg. Moreover, by the Hirzebruch signature theorem c1 (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 + ≤ 2 e + 3 σ + 5 {\displaystyle b_{+}\leq 2e+3\sigma +5}

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

b − + 4 b 1 ≤ 4 b + + 9. {\displaystyle b_{-}+4b_{1}\leq 4b_{+}+9.}

Combining the Noether inequality with the Noether formula 12χ=c1+c2 gives

5 c 1 ( X ) 2 − c 2 ( X ) + 36 ≥ 12 q {\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:

5 c 1 ( X ) 2 − c 2 ( X ) + 36 ≥ 0 ( c 1 2 ( X )  even ) {\displaystyle 5c_{1}(X)^{2}-c_{2}(X)+36\geq 0\quad (c_{1}^{2}(X){\text{ even}})}
5 c 1 ( X ) 2 − c 2 ( X ) + 30 ≥ 0 ( c 1 2 ( X )  odd ) . {\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 K > 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 → H 0 ( O X ) → H 0 ( K ) → H 0 ( K | D ) → H 1 ( O X ) → {\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 p g − 1 ≤ h 0 ( K | D ) . {\displaystyle p_{g}-1\leq h^{0}(K|_{D}).}

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

h 0 ( K | D ) − 1 ≤ 1 2 deg D ⁡ ( K ) = 1 2 K 2 . {\displaystyle h^{0}(K|_{D})-1\leq {\frac {1}{2}}\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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