Noether inequality [SEP] named after
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" = −"c"("X"),
) = 2"e" + 3"σ", where "e" = "c"("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
or equivalently using "e" = 2 – 2 "b" + "b" + "b"
Combining the Noether inequality with the Noether formula 12χ="c"+"c" gives
where "q" is the irregularity