Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHolomorphic curves for Legendrian surgery
83
0
0.0
(
0
)
Authors
Tobias Ekholm
Ask ChatGPT about the research
No Arabic abstract
Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $Lambdasubset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $Lambda$ we get a Weinstein cobordism $X_{Lambda}$ with a collection of Lagrangian co-core disks $C$ corresponding to $Lambda$. In cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{ast}(C)$ of $C$ in the Weinstein manifold $X_{Lambda}=Xcup X_{Lambda}$is naturally isomorphic to the Legendrian differential graded algebra $CE^{ast}(Lambda)$ of $Lambda$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which were only sketched. The purpose of this paper is to provide proofs of these properties.
rate research
Read More
We determine the skein-valued Gromov-Witten partition function for a single toric Lagrangian brane in $mathbb{C}^3$ or the resolved conifold. We first show geometrically they must satisfy a certain skein-theoretic recursion, and then solve this equation. The recursion is a skein-valued quantization of the equation of the mirror curve. The solution is the expected hook-content formula.
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.
Given a symplectic three-fold $(M,omega)$ we show that for a generic almost complex structure $J$ which is compatible with $omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $ggeq 0$ representing a homology class $beta$ in $H_2(M,Z)$ with $c_1(M).beta=0$, provided that the divisibility of $beta$ is at most 4 (i.e. if $beta=nalpha$ with $alphain H_2(M,Z)$ and $nin Z$ then $nleq 4$). Moreover, each such curve is embedded and 4-rigid.
We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by neck-stretching and splashing.
In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
