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 , and extended to higher dimensions by . It is a complex analog of Morse theory, which 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 -dimensional projective complex manifold to the projective line P¹. Also suppose that all critical points of f are non-degenerate and have distinct images x₁,...,x_n in P¹. Pick any other point x in P¹. The fundamental group — which is the free group F_n-1 on n-1 generators — 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 Y_x = f^-1(x) at x. Note that this is the middle homology since the fibre has complex dimension k, hence real dimension 2k. The monodromy action of 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 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.

Example

Consider the projective family of hyperelliptic curves of genus defined by

where is the parameter and . Then, this family has double-point degenerations whenever . Since the curve is a connected sum of tori, the intersection form on of a generic curve is the matrix

we can easily compute the Picard-Lefschetz formula around a degeneration on . Suppose that are the -cycles from the -th torus. Then, the Picard-Lefschetz formula reads

if the -th torus contains the vanishing cycle. Otherwise it is the identity map.

See also

• Lefschetz pencil

References