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.
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 For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold.
In particular, we may assume there is an effective divisor D representing K. We then have an exact sequence so
is a special divisor and the Clifford inequality applies, which gives 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.