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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'' = −''c''1(''X''), and let ''p''g = ''h''0(''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. 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 + 2''p''g. Moreover by the Hirzebruch signature theorem ''c''12 (''X'') = 2''e'' + 3''σ'', where ''e'' = ''c''2(''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''1 + ''b''+ + ''b''- : Combining the Noether inequality with the Noether formula 12χ=''c''12+''c''2 gives : where ''q'' is the irregularity of a surface, which leads to a slightly weaker inequality, which is also often called the Noether inequality: : : Surfaces where equality holds (i.e. on the Noether line) are called Horikawa surfaces. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Noether inequality」の詳細全文を読む スポンサード リンク
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