Weinstein manifolds revisited

Abstract

This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds. Weinstein manifolds, domains, cobordismsWe begin with a notion of a Liouville domain. Let (X, ω) be a 2n-dimensional compact symplectic manifold with boundary equipped with an exact symplectic form ω. A Liouville structure on (X, ω) is a choice of a primitive λ, dλ = ω, called Liouville form such that λ| ∂X is a contact form and the orientation of ∂X by the form λ ∧ dλ n−1 | ∂X coincides with its orientation as the boundary of symplectic manifold (X, ω). The vector field Z, that is ω-dual to λ, i.e. ι(Z)ω = λ, is also called Liouville. It satisfies the condition L Z ω = ω which means that its flow is conformally symplectically expanding. The contact boundary condition is equivalent to the outward transversality of Z to ∂X. A Liouville domain X can always be completed to a Liouville manifold X by attaching a cylindrical end:and extending λ to X as equal to e s (λ| ∂X ) on the attached end. We will be constantly going back and forth between these two tightly related notions of Liouville domains and Liouville manifolds.Given a Liouville structure L = (X, ω, Z) we say that a Liouville structure L = (X , ω, Z) is obtained by a radial deformation from L if there exists a function h : *

Covariantly functorial wrapped Floer theory on Liouville sectors

“…Similarly, a sutured Liouville manifold is a Liouville manifoldX together with a codimension one submanifold-with-boundary F 0 ⊆ ∂ ∞X and a contact form λ defined over Nbd F 0 such that (F 0 , λ) is a Liouville domain. (Compare with the notion of a "Weinstein pair" from [16].) Given a sutured Liouville domain (X 0 , F 0 ), the Reeb vector field of λ is transverse to F 0 since dλ| F 0 is symplectic, and thus determines a local coordinate chart F 0 × R |t|≤ε → ∂X 0 in which the contact form λ equals dt + λ| F 0 .…”

“…Definition 2. 16. An open book decomposition of a contact manifold Y consists of a binding B ⊆ Y (a codimension two submanifold), a tubular neighborhood B ×D 2 ⊆ Y , a submersion π : Y \ B → S 1 standard over B × D 2 , and a contact form α on Y such that the pages of the open book (π −1 (a), dα) are symplectic, and α = (1 + 1…”

Simplicial descent for Chekanov–Eliashberg dg‐algebras

“…More precisely, the only Reeb chords and holomorphic disks that exist are confined to building blocks in $$X$$ associated to $$\sigma _k$$ and its subfaces. (3)Equivalently, stopping $$X$$ away from $$\sigma _k$$ can be viewed as a form of completion of $$V^{(k)}_{\sigma _k} \times {\mathbb {R}}^{2k}$$ stopped at the Weinstein hypersurface $$(\bigsqcup _{f\in F} \# \bm{V}_{\supsetneq \sigma _k,f}) \times {\mathbb {R}}^{2k}$$ by negative ends. This is in spirit similar to convexification as in [24, Remark 2.30] and more precisely described in [20, figure 2.1 and Remark 2.10].…”

“…Roughly speaking, using each Weinstein manifold $$V \in \bm{V}$$ we construct a simplicial handle , such that when glued together according to $$C$$ gives $$X$$ up to Weinstein isomorphism, see Subsection 1.3 for a more precise description. The definition of a simplicial decomposition generalizes the notion of Weinstein connected sum [2, 4, 20] and if we allow for Weinstein cobordisms with nonempty negative ends, simplicial decompositions may be used to give a surgery description of stopped Weinstein manifolds used in the study of partially wrapped Fukaya categories [1, 3, 17, 24, 40].…”

Arboreal singularities in Weinstein skeleta

“…We review here basic definitions of Weinstein manifolds. A more in depth discussion of Weinstein manifolds and their context in symplectic geometry can be found in [EG91], [CE12], and [Eli17].…”