Picard–Lefschetz theory

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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 Picard & Simart (1897), and extended to higher dimensions by Lefschetz (1924). 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. Deligne & Katz (1973) 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[edit]

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 x1,...,xn in P1. Pick any other point x in P1. The fundamental group π1(P1 – {x1, ..., xn}, x) is generated by loops wi going around the points xi, and to each point xi there is a vanishing cycle in the homology Hk–1(Yx) of the fiber at x. There is a monodromy action of π1(P1 – {x1, ..., xn}, x) on Hk–1(Yx), described as follows by the Picard–Lefschetz formula. (The action of monodromy on other homology groups is trivial.) The monodromy action of a generator wi of the fundamental group on \gamma ∈ Hk–1(Yx) is given by

w_i(\gamma) = \gamma+(-1)^{k(k+1)/2}\langle \gamma,\delta_i\rangle \delta_i

where δi is the vanishing cycle of xi. This formula appears implicitly for n = 2 (without the explicit coefficients of the vanishing cycles δi) in Picard & Simart (1897, p.95). Lefschetz (1924, chapters II, V) gave the explicit formula in all dimensions.