Picard–Lefschetz theory

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.

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,...,x_n in P¹. Pick any other point x in P¹. The fundamental group π₁(P¹ – {x₁, ..., x_n}, 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 – {x₁, ..., x_n}, 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 ${\displaystyle \gamma }$ ∈ H_k(Y_x) is given by

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