Mathematics

For a Legendrian link Λ⊂ J 1 M with M=R or S 1 , immersed exact Lagrangian fillings L⊂Symp( J 1 M)≅ T ∗ ( R >0 ×M) of Λ can be lifted to conical Legendrian fillings Σ⊂ J 1 ( R >0 ×M) of Λ . When Σ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation α:A(Σ)→Z/2 of the LCH algebra of Σ , there is an induced augmentation ϵ (Σ,α) :A(Λ)→Z/2 . With Σ fixed, the set of homotopy classes of all such induced augmentations, I Σ ⊂Aug(Λ)/∼ , is a Legendrian isotopy invariant of Σ . We establish methods to compute I Σ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary n≥1 , we give examples of Legendrian torus knots with 2n distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when ρ≠1 and Λ⊂ J 1 R every ρ -graded augmentation of Λ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of ρ -graded augmented Legendrian cobordism.

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the associated Chekanov-Eliashberg differential graded algebra, and ruling polynomial via combinatorics of the decompositions of the associated front projection.

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital A ∞ -category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove "augmentations are sheaves" in the singular case.

We prove that the augmentation variety of any positive braid Legendrian link carries a natural cluster K 2 structure. We present an algorithm to calculate the cluster seeds that correspond to the admissible Lagrangian fillings of the positive braid Legendrian links. Utilizing augmentations and cluster algebras, we develop a new framework to distinguish exact Lagrangian fillings.

The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture.

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated version of the Carrell-Liberman theorem. As an immediate consequence, this confirms a conjecture raised by Battaglia and Zaffran on the basic Hodge numbers of symplectic toric quasifolds. As an aside, we also present a symplectic approach to the calculation of basic Betti numbers of symplectic toric quasifolds.

We construct a Kodaira-Spencer map from the big quantum cohomology of a sphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg potential function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangian and we show that Kodaira-Spencer map is a ring isomorphism.

We present an algebraic framework for the computation of low-degree cohomology of a class of bigraded complexes which arise in Poisson geometry around (pre)symplectic leaves. We also show that this framework can be applied to the more general context of Lie algebroids. Finally, we apply our results to compute the low-degree cohomology in some particular cases.

Let N be a closed manifold and U⊂ T ∗ (N) a bounded domain in the cotangent bundle of N , containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians L 0 , L 1 , which depends linearly on the boundary depth of the Floer complexes of ( L 0 ,F) and ( L 1 ,F) , where F is a fiber of the cotangent bundle.

Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on V× T 2 . We prove that all such structures are universally tight in dimension 5 , independent on whether the original contact manifold is tight or overtwisted. In arbitrary dimensions, we give very restrictive obstructions to the existence of strong symplectic fillings of Bourgeois manifolds, from which we obtain broad families of new examples of weakly but not strongly fillable contact 5 --manifolds, and the first examples of weakly but not strongly fillable contact structures in all odd dimensions. In particular, we answer negatively a question of Lisi--Marinković--Niederkrüger concerning the strong fillability of the Bourgeois contact manifold associated to the open book with monodromy given by a single Dehn--Seidel twist on the unit cotangent bundle of the n -sphere. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n -torus has a unique symplectically aspherical strong filling up to diffeomorphism.