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
\( {\displaystyle p_{g}\leq {\frac {1}{2}}c_{1}(X)^{2}+2.} \)
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 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
5 c 1 ( X ) 2 − c 2 ( X ) + 36 ≥ 12 q {\displaystyle 5c_{1}(X)^{2}-c_{2}(X)+36\geq 12q} {\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 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
\( {\displaystyle 0\to H^{0}({\mathcal {O}}_{X})\to H^{0}(K)\to H^{0}(K|_{D})\to H^{1}({\mathcal {O}}_{X})\to } \)
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
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