Positive arborealization of polarized Weinstein manifolds

Álvarez-Gavela, Daniel; Eliashberg, Yakov; Nadler, David A.

doi:10.48550/arxiv.2011.08962

Cited by 7 publications

(22 citation statements)

References 8 publications

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“…In this Section we will prove the h-principle for step-2 distributions (Theorem 1.5) and its corollary about the classification of (3,5) and (3,6) distributions (Theorem 1.7). The proof can be found in Subsection 8.4.…”

Section: Flexibility Of Step-2 Distributionsmentioning

confidence: 98%

“…In both cases, the main point is that, due to the presence of discontinuities, there is no formal data associated to the objects we consider. h-Principles without homotopical assumptions play now a central role in Symplectic and Contact Topology through the arborealisation programme [35,1,2,3].…”

Section: Jiggling For Principal Coversmentioning

confidence: 99%

“…On the other hand, Adachi non-integrability is stronger than being of step 2. The two notions coincide for (3,5) and (3,6). In particular, his claims should recover our step-2 result (Theorem 1.5) for odd rank.…”

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confidence: 94%

“…The remaining cases are: Corank-1 (which are even-contact structures), rank-3 (classified by Theorem 1.7) and rank-2 (the so-called (2,3,5,6) structures, for which nothing is known). Beyond dimension 6, we propose, jointly with I. Zelenko, the following concrete conjecture.…”

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confidence: 99%

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Convex integration with avoidance and hyperbolic (4,6) distributions

Martínez-Aguinaga¹,

Pino²

2021

Preprint

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This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3, 5) and (3, 6) distributions follows as a particular case.We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration.

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Section: Flexibility Of Step-2 Distributionsmentioning

confidence: 98%

Section: Jiggling For Principal Coversmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 99%

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Convex integration with avoidance and hyperbolic (4,6) distributions

Martínez-Aguinaga¹,

Pino²

2021

Preprint

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“…Let X be a 2n-dimensional Weinstein manifold with ideal contact boundary ∂X, and let (V, λ) be a (2n − 2)-dimensional Weinstein manifold. A Weinstein hypersurface is a Weinstein embedding (V, λ V := λ| V ) → (X \ Skel X, λ) such that the induced map V −→ ∂X is an embedding, see [Avd12,Eli18,Syl19,GPS20,ÁGEN20]. Throughout this paper we use the notation V → ∂X to denote a Weinstein hypersurface.…”

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confidence: 99%

Simplicial descent for Chekanov-Eliashberg dg-algebras

Asplund¹

2021

Preprint

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We introduce a type of surgery decomposition of Weinstein manifolds we call simplicial decompositions. The main result of this paper is that the Chekanov-Eliashberg dg-algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and we show that there is a one-to-one correspondence (up to Weinstein homotopy) between simplicial decompositions and so-called good sectorial covers. As an application we explicitly compute the Chekanov-Eliashberg dg-algebra of the Legendrian attaching spheres of a plumbing of copies of cotangent bundles of spheres of dimension at least three according to any plumbing quiver. We show by explicit computation that this Chekanov-Eliashberg dg-algebra is quasi-isomorphic to the Ginzburg dg-algebra of the plumbing quiver. Contents 19 3.1. Simplicial decompositions and good sectorial covers 19 3.2. Relation to sectorial descent 26 4. Applications 27 4.1. Legendrian submanifolds-with-boundary-and-corners 27 4.2. The Chekanov-Eliashberg cosheaf 30 4.3. Plumbing of cotangent bundles of spheres 31 References 36

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On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

Bai¹,

Côté²

2023

Compositio Math.

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We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$ . As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

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Positive arborealization of polarized Weinstein manifolds

Cited by 7 publications

References 8 publications

Convex integration with avoidance and hyperbolic (4,6) distributions

Convex integration with avoidance and hyperbolic (4,6) distributions

Simplicial descent for Chekanov-Eliashberg dg-algebras

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

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