Picard–Lefschetz theory

Summary

Picard–Lefschetz theory studies the topology of a complex manifold through a holomorphic function on it with nondegenerate critical points. It looks at how the fibres change near the critical values. Its central result, the Picard–Lefschetz formula, describes the monodromy around a critical value in terms of the vanishing cycle $\delta$ of the critical point. Let $X_t$ be a smooth fibre of complex dimension $n$, near a critical value with a single nondegenerate critical point. Let $T$ be the monodromy on $H_n(X_t; \Z)$ around this value. Then $T(a) = a + \varepsilon_n \langle a, \delta \rangle \delta$, where $\langle \cdot, \cdot \rangle$ is the intersection form. Here $\varepsilon_n = \pm 1$ is a sign that depends only on $n$ and on orientation conventions. Émile Picard developed the case of surfaces and Solomon Lefschetz the general case. Lefschetz used it to study the topology of algebraic varieties through Lefschetz pencils, together with the Lefschetz hyperplane theorem. Its version in étale cohomology was a main tool in Deligne's proof of the Weil conjectures.