Bulletin (new Series) of the American Mathematical Society

One route to investigating the differential topology of real manifolds is through Morse theory. A smooth manifold M is decomposed into the level-sets f−1(x) of a smooth real-valued function f : M → R, and the global topology of M emerges from the descriptions of how these level-sets change. One can understand the whole homotopy type of M from this point of view [Mil69], or one can pass more quickly to algebra by defining the Morse complex, which has generators the critical points {x ∈ M | df(x) = 0} of f and a differential counting downward gradient flow-lines γ : R → M which solve γ′(t) = −∇f(γ(t)). This Morse complex computes the usual singular cohomology, and one immediate consequence is a nontrivial lower bound on the number of critical points of a generic smooth function on a closed manifold. For complex algebraic varieties, Picard-Lefschetz theory [La81] is a complexification of this picture, which studies a projective manifold through the level-sets of a holomorphic function f : X C (since holomorphic functions on compact complex manifolds are constant, this will be defined only away from some subvariety). The locus of critical values being finite in C and having connected complement, the inverse images of regular points f−1(x) will typically all be diffeomorphic; the global topology of X now emerges through the monodromy, which describes how these fibres are twisted globally in the family. Picard-Lefschetz theory provides a kind of dimensional-induction machine, in which one studies varieties through their hyperplane sections which, being of lower dimension, are presumed more tractable. In a short but influential note [Arn95], V. I. Arnol’d pointed out that the monodromy transformations of Picard-Lefschetz theory have a basic connection to symplectic topology. The heart of this connection is straightforward to describe. Morse functions have isolated singularities modeled locally on nondegenerate quadratic forms. Over C, the unique local model is the map π : (z1, . . . , zn) → ∑ j z 2 j . The general fibre π−1(1) ⊂ C is symplectically diffeomorphic to the cotangent bundle of a sphere T ∗Sn−1, which carries the canonical symplectic structure of classical mechanics, and Arnol’d’s basic observation is that the monodromy of the family of varieties {π−1(t) | t ∈ S} is a symplectic diffeomorphism of T ∗Sn−1. In the lowest nontrivial dimension n = 2, the general fibre is an annulus, the monodromy map is a Dehn twist in the obvious waist curve {(z1, z2) ∈ R | z 1 + z 2 = 1}, and the symplectic property asserts that this preserves area. In higher dimensions there is an analogous Dehn twist, which acts antipodally on the corresponding real sphere (i.e., the zero-section of T ∗Sn−1) and is compactly supported in a neighbourhood of that sphere. Even in the local model, the symplectic structure keeps track of information not visible in classical differential topology: when n = 3, the Dehn twist on an affine quadric surface is of infinite order as a compactly supported symplectomorphism, but its square is differentiably isotopic to the identity. It has become clear over the last decade that this change in perspective, from algebraic to symplectic topology in the context of Picard-Lefschetz theory, is actually rather profound, providing a key inroad into the symplectic natures of algebraic