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In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book , and extended to higher dimensions by . It is a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures. ==Picard–Lefschetz formula== The Picard–Lefschetz formula describes the monodromy at a critical point. Suppose that ''f'' is a holomorphic map from an ''(k+1)''-dimensional projective complex manifold to the projective line P1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images ''x''1,...,''x''''n'' in P1. Pick any other point ''x'' in P1. The fundamental group π1(P1 – , ''x'') is generated by loops ''w''''i'' going around the points ''x''''i'', and to each point ''x''''i'' there is a vanishing cycle in the homology ''H''''k''(''Y''''x'') of the fiber at ''x''. Note that this is the middle homology since the fibre has complex dimension ''k'', hence real dimension ''2k''. The monodromy action of π1(P1 – , ''x'') on ''H''''k''(''Y''''x'') is described as follows by the Picard–Lefschetz formula. (The action of monodromy on other homology groups is trivial.) The monodromy action of a generator ''w''''i'' of the fundamental group on '''' ∈ ''H''''k''(''Y''''x'') is given by : where δ''i'' is the vanishing cycle of ''x''''i''. This formula appears implicitly for ''k'' = 2 (without the explicit coefficients of the vanishing cycles δ''i'') in . gave the explicit formula in all dimensions. 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「Picard–Lefschetz theory」の詳細全文を読む スポンサード リンク
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