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In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety ''X'' embedded in projective space and a hyperplane section ''Y'', the homology, cohomology, and homotopy groups of ''X'' determine those of ''Y''. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. == The Lefschetz hyperplane theorem for complex projective varieties == Let ''X'' be an ''n''-dimensional complex projective algebraic variety in CP''N'', and let ''Y'' be a hyperplane section of ''X'' such that ''U'' = ''X'' ∖ ''Y'' is smooth. The Lefschetz theorem refers to any of the following statements: # The natural map ''H''''k''(''Y'', Z) → ''H''''k''(''X'', Z) in singular homology is an isomorphism for ''k'' < ''n'' − 1 and is surjective for ''k'' = ''n'' − 1. # The natural map ''H''''k''(''X'', Z) → ''H''''k''(''Y'', Z) in singular cohomology is an isomorphism for ''k'' < ''n'' − 1 and is injective for ''k'' = ''n'' − 1. # The natural map π''k''(''Y'', Z) → π''k''(''X'', Z) is an isomorphism for ''k'' < ''n'' − 1 and is surjective for ''k'' = ''n'' − 1. Using a long exact sequence, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are: # The relative singular homology groups ''H''''k''(''X'', ''Y'', Z) are zero for . # The relative singular cohomology groups ''H''''k''(''X'', ''Y'', Z) are zero for . # The relative homotopy groups π''k''(''X'', ''Y'') are zero for . 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「Lefschetz hyperplane theorem」の詳細全文を読む スポンサード リンク
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