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Zentralblatt MATH identifier Subjects Primary: 53D Contact manifolds, general Secondary: 53D Almost contact and almost symplectic manifolds 57R Symplectic and contact topology. Keywords Legendrian handlebody Lefschetz fibrations Weinstein structure h-principle. Casals, Roger; Murphy, Emmy. Legendrian fronts for affine varieties.

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A microlocal category associated to a symplectic manifold

Abstract Article info and citation First page References Abstract In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. Article information Source Duke Math.

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Export citation. Export Cancel. References [1] M. Abouzaid, A geometric criterion for generating the Fukaya category , Publ. Even better, the statement of Theorem 1. Note that each of the three conclusions addresses an important aspect of the symplectic topology of these manifolds, and the three of them are direct consequences of the understanding of Legendrian fronts.

In the complex 3 —dimensional case, Theorem 1. Theorem 1. Computation of pseudoholomorphic invariants, such as symplectic homology and, more generally, a model for the wrapped Fukaya category. These invariants can be extracted from the Legendrian contact differential graded algebra [ 20 , 28 ] via the generation criterion [ 1 ] and the surgery exact sequences [ 11 , 12 ]. Finding exact Lagrangian submanifolds in Weinstein manifolds.

Given the Legendrian handlebody decomposition of a Weinstein manifold we can use combinatorial arguments to construct Lagrangian cobordisms inside the Weinstein manifold [ 3 , Section 3. Detection of flexibility [ 19 ] and subflexibility [ 57 ] of a Weinstein manifold. These properties are defined in terms of Weinstein handlebodies which we explicitly produce, from which the properties can be directly verified. In the case of flexibility, the h -principle produces novel symplectomorphisms between a priori distinct Weinstein manifolds. To these three applications, we can add the more familiar arguments from Kirby calculus to the Weinstein setting such as deciding whether two given Weinstein manifolds are deformation equivalent.

The front dictionary transforms this into a combinatorial problem of Legendrian isotopies and moves in the front projection, which is potentially more manageable. The systematic computation of the Legendrian differential graded algebra in Application A is not a simple task. In general, the computation requires T.

Theorems 1. In contrast, Theorem 4. This article is also an open invitation to hunt for Lagrangian submanifolds in affine manifolds: for instance, we suggest hunting in the family of affine hypersurfaces presented in Subsection 2.

Subsection 3. The following theorem is another example of Application B, which is proved in Section 4. In this line, algebraic geometers have been proving a wealth of significant results on the algebraic isomorphism types of affine complex varieties [ 26 , 59 , 73 ] but the underlying Stein structures are not quite understood; see [ 54 , 55 , 66 ] for progress in this direction. Being affine hypersurfaces, the results of the present article provide a systematic way of studying these Milnor fibers from the symplectic viewpoint. In order to continue the study of the symplectic topology of affine manifolds, we focus on the most salient instance of an exotic affine manifold: the Koras—Russell cubic.

It is the affine cubic hypersurface defined by.


As an application of the Legendrian front dictionary we will prove the following theorem:. This opens the way to the study of affine varieties up to deformation equivalence via the study of explicit Legendrian links, and we encourage the readers to study the underlying Stein structures of many other interesting affine algebraic varieties [ 73 ]. In particular, the symplectic topology of acyclic surfaces, which started with the article [ 66 ] , and other instances of exotic affine C 3 , can be the subject of exciting future work.

Third, Theorems 1. In order to illustrate this, we present the following theorem :. In addition, the Weinstein 4 —manifolds X and Y both contain exact Lagrangian tori. Both symplectic manifolds X and Y have been studied prominently in the literature, and the statement of Theorem 1. But as promised, the method of proof we present in this article is genuinely different from these techniques, and it provides an understanding of the mirror symmetry correspondence from the Legendrian viewpoint.

Finally, the article also contains material discussing higher—dimensional Reidemeister moves in Subsection 2. These constitute foundational material in the study of higher—dimensional Weinstein structures and the contact topology of their boundaries. The arc of the work is organized as follows: Section 2 contains the material related to Legendrian front projections, Dehn twists, Legendrian isotopies, and loose charts.

Then Section 3 presents the basic material on Lefschetz bifibrations broadening the range of applications of Recipe 3.

Furthermore, every Weinstein structure is realized in this way, unique up to Stein deformation [ 19 ]. Throughout this article we will freely pass from one perspective to the other. Acknowledgements : We are grateful to Y. Eliashberg, A. Keating, M. McLean, O. Plamenevskaya and K. Siegel for valuable discussions and their interest in this work.

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Special thanks go to L. Starkston and U. In this section we introduce an algorithm for drawing fronts of Lagrangians acted on by Dehn twists, and explain the basic rules for the diagramatic calculus in the Legendrian front. These combinatorial rules constitute a major portion of Recipe 3.


Each of the first six subsections contributes with an ingredient leading up to the Legendrian stacking, presented in Subsection 2. Then the second part of Subsection 2. These first six subsections can be shortly described as follows: Subsection 2. Let us start with the first building block of a Weinstein manifold, its subcritical skeleton.

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The subcritical smooth topology of a Weinstein manifold does not contain meaningful symplectic topology information; this is illustrated by M. Cieliebak splitting principle [ 19 ] for subcritical Weinstein manifolds. The subcritical topology of a Weinstein manifold can be quite arbitrary, and thus we will restrict ourselves to simple subcritical skeleta for pictorial purposes; the dictionary works with arbitrary Weinstein manifolds as long as there is an efficient manner to depict their subcritical topology.

The data determining the subcritical Weinstein manifold W 0 is. This discussion will be expanded in Sections 2. Before this, we discuss the main theorems about loose Legendrian embeddings which are needed in order to build Recipe 3.

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In this section we discuss loose Legendrians embeddings [ 56 ].