We show that the Gromov width of the Grassmannian of complex k-planes in C^n is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general results. For example, if a compact manifold N with an integral symplectic form omega admits a Hamiltonian circle action with a fixed point p such that all the isotropy weights at p are equal to one, then the Gromov width of (N,omega) is at least one. We use holomorphic techniques to prove the upper bound.
We find an upper bound for the Gromov width of coadjoint orbits of compact Lie groups with respect to the Kirillov Kostant Souriau form by computing certain Gromov Witten invariants, the approach presented here is closely related to the one used by Gromov in his celebrated nonsqueezing theorem.
We compute the Gromov width of homogeneous Kaehler manifolds with second Betti number equal to one. Our result is based on the recent preprint [4] and on the upper bound of the Gromov width for such manifolds obtained in [6].
We find an upper bound for the Gromov width of coadjoint orbits of U(n) with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated non-squeezing theorem.
Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_lambda through lambda in Lie(T)^* is canonically a symplectic manifold. Therefore we can ask the question about its Gromov width. In many known cases the Gromov width is exactly the minimum over the set {< alpha_j^{\vee},lambda > ; alpha_j^{\vee} a coroot and < alpha_j^{\vee},lambda > positive}. We show that the Gromov width of coadjoint orbits of the unitary group and of most of the coadjoint orbits of the special orthogonal group is at least the above minimum. The proof uses the torus action coming from t…
By Delzant's theorem, closed symplectic toric manifolds are classified by the images of moment maps. In the case of a generalized Bott manifold, this image is a polytope $P$ combinatorially equivalent to the product of simplices. We compute the Gromov width of generalized Bott manifolds in terms of the defining inequalities of $P$.
Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_λ through λin the dual of the Lie algebra of T, is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width. In many known cases the width is exactly the minimum over the positive results of pairing λwith coroots: min{< α_j^{\vee},λ> ; α_j^{\vee} is a coroot and < α_j^{\vee},λ> is positive}. We will show that the Gromov width for regular coadjoint orbits of the special orthogonal group is at least this minimum. The proof uses the torus action coming from the Gelfand-Tsetlin s…
We prove a conjecture of Barraud-Cornea for orientable Lagrangian surfaces. As a corollary, we obtain that displaceable Lagrangian 2--tori have finite Gromov width. In order to do so, we adapt the pearl complex of Biran-Cornea to the non-monotone situation based on index restrictions for holomorphic discs.
For generic $r=(r_1,\ldots,r_n) \in \mathbb{R}^n_+$ the space $\mathcal{M}(r)$ of $n$--gons in $\mathbb{R}^3$ with edges of lengths $r$ is a smooth, symplectic manifold. We investigate its Gromov width and prove that the expression $$2π\min \{2 r_j, (\sum_{i \neq j} r_i) - r_j\,\,|\, j=1,\ldots,n\}$$ is the Gromov width of all (smooth) $5$--gon spaces and of $6$--gon spaces, under some condition on $r \in \mathbb{R}^6_+$. The same formula constitutes a lower bound for all (smooth) spaces of $6$--gons. Moreover, we prove that the Gromov width of $\mathcal{M}(r)$ is given by the above expression…
We prove that the Gromov width of coadjoint orbits of the symplectic group is at least equal to the upper bound known from the works of Zoghi and Caviedes. This establishes the actual Gromov width. Our work relies on a toric degeneration of a coadjoint orbit to a toric variety. The polytope associated to this toric variety is a string polytope arising from a string parametrization of elements of a crystal basis for a certain representation of the symplectic group.
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic u…
We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of the 4-torus with its given linear symplectic form.
We use the Gelfand-Tsetlin pattern to construct an effective Hamiltonian, completely integrable action of a torus T on an open dense subset of a coadjoint orbit of the unitary group. We then identify a proper Hamiltonian T-manifold centered around a point in the dual of the Lie algebra of T. A theorem of Karshon and Tolman says that such a manifold is equivariantly symplectomorphic to a particular subset of R^2D. This fact enables us to construct symplectic embeddings of balls into certain coadjoint orbits of the unitary group, and therefore obtain a lower bound for their Gromov width. Using t…
We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of $J$-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of any given symplectically embedded ball. The areas of these disks --- and hence the sizes of these balls --- are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit co…
We give an explicit formula for the Gromov width for a class of symplectic toric manifolds constructed from simple graphs. As a corollary, we show a version of non-squeezing theorem with respect to the inclusion of connected graphs.
We give a uniform proof for the conjectured Gromov width of coadjoint orbits of all compact connected simple Lie groups, by analyzing simplices in Newton-Okounkov bodies.
We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds $(M, ω)$ with $b_2(M)=1$. As an application we obtain an upper bound on the Seshadri constant $ε(L)$ where $L$ is the ample line bundle on $M$ such that $c_1(L)=[\fracωπ]$.
The complex projective plane CP^2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of CP^2, meaning that pinwheels are Lagrangian barriers in the sense of Biran. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below 3.
We prove that the Gromov width of any Bott-Samelson variety birational to a Schubert variety and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.
In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness cons…
Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.
We construct a Gelfand-Zeitlin system on a one-parameter family of $G_2$ coadjoint orbits that are multiplicity-free Hamiltonian $SU(3)$-spaces. Using this system we prove a lower bound for the Gromov width of these orbits. This lower bound agrees with the known upper bound.
A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. Harada and Kaveh described how one incorporates a symplectic structure into this process, providing a very useful tool for solving certain problems in symplectic geometry. Below we present applications of this method to questions about the Gromov width, and cohomological rigidity problems.
Let $X$ be an $n$-dimensional smooth complex projective variety embedded in $\mathbb{C}\mathbb{P}^{N}$. We construct a smooth family $\mathcal{X}$ over $\mathbb{C}$ with an embedding in $\mathbb{C}\mathbb{P}^{N} \times \mathbb{C}$ whose generic fiber is $X$ and the special fiber is the torus $(\mathbb{C}^*)^n$ sitting in $\mathbb{C}\mathbb{P}^{N}$ via a monomial embedding. We use this to show that if $ω$ is an integral Kähler form on $X$ then for any $ε> 0$ there is an open subset $U_ε\subset X$ such that $vol(X \setminus U_ε) < ε$ and $U_ε$ is symplectomorphic to $(\mathbb{C}^*)^n$ equipped w…
