Picard–Lefschetz theory and dilating ℂ*-actions

Abstract

Abstract. We consider C * -actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation. IntroductionIt has been gradually recognized that certain classes of non-closed symplectic manifolds admit symmetries of a new kind. These symmetries are not given by groups acting on the manifold, but instead appear as extra structure on Floer cohomology, or more properly on the Fukaya category. The first example may have been the bigrading on the Floer cohomology of certain Lagrangian spheres in the Milnor fibres of type (A) hypersurface singularities in C n , described in [12] (it turned out later [21, Section 20] that this is compatible with the A ∞ -structure of the Fukaya category only if n ≥ 3). The geometric origin of such symmetries (on the infinitesimal level) has been studied in [25], with applications in [24,15,1]. A roadmap is provided (via mirror symmetry) by the theory of equivariant coherent sheaves, and its applications in algebraic geometry and geometric representation theory; the relevant literature is too vast to survey properly, but [18,8] have been influential for the developments presented here.In [22], the example from [12] was used as a test case for talking about such symmetries in an algebraic language of A ∞ -categories with C * -actions. Here, we generalize that approach, and combine it with the symplectic version of Picard-Lefschetz theory [21]. Recall that classical Picard-Lefschetz theory provides (among other things) a way of computing the intersection pairing in the middle-dimensional homology of an affine algebraic variety, by induction on the dimension. As one consequence of our construction, one gets a similar machinery for defining and computing algebraic analogues of the q-intersection numbers from [25]. Aside from their intrinsic interest, these q-intersection numbers have implications for the topology of Lagrangian submanifolds. These are similar in spirit to those derived in [24], but benefit from the more rigid setup of C * -actions, as well as the greater ease of doing computations in a purely algebraic framework. In particular, Example 1.20 would be out of reach of the methods in [24], since those only involved the first derivative in the equivariant parameter q around q = 1, whereas (1.80) vanishes at least to order 2 at that point. Another noteworthy comparison is [14], which contains an example of a

Connections on equivariant Hamiltonian Floer cohomology

“…Cyclic symmetries exist for a much wider class of (not necessarily Hamiltonian) symplectic Floer cohomology groups. Our main point of reference is [27], which only considers the case p = 2; hence, we will ultimately restrict to that case, even though this can be a bit misleading.…”

Legendrian fronts for affine varieties

“…The two following subsections suffice for our purpose. The general technique of working with Lefschetz bifibrations has been used in the literature for a number of years, see [5,9,51,52,63,64,65]. The reader familiar with [63, Part III] can move directly to Subsection 3.3.…”

“…These Weinstein manifolds (M n b , λ, ϕ) are a variation on the Weinstein manifolds (X n 1,b , λ, ϕ), and first appear in the work of P. Seidel [65], to whom we are grateful for useful discussions on the symplectic topology of these manifolds and their relation to mirror symmetry. The variation consists in modifying the defining polynomial by changing the degree-0 constant 1 to a linear term on x.…”

A simplicial construction of G‐equivariant Floer homology

“…These and later Floer‐type invariants are often enriched by various kinds of symmetries. For example, Seiberg–Witten Floer homology and cylindrical contact homology are intrinsically ‐equivariant theories [4, 23, 27], and Fukaya categories often carry actions of mapping class groups (for example, [22, 37]) and Lie algebras (for example, [8, 35]).…”