An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase.
This work has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic 4-manifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point a_0, given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at a_0 must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that we show is powerful.
In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by reducing our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.
We study the symplectic embedding capacity function $C_β$ for ellipsoids $E(1,α)\subset R^4$ into dilates of polydisks $P(1,β)$ as both $α$ and $β$ vary through $[1,\infty)$. For $β=1$ Frenkel and Mueller showed that $C_β$ has an infinite staircase accumulating at $α=3+2\sqrt{2}$, while for integer $β\geq 2$ Cristofaro-Gardiner, Frenkel, and Schlenk found that no infinite staircase arises. We show that, for arbitrary $β\in (1,\infty)$, the restriction of $C_β$ to $[1,3+2\sqrt{2}]$ is determined entirely by the obstructions from Frenkel and Mueller's work, leading $C_β$ on this interval to have…
We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic manifolds. As an application, we show that any toric symplectic stack can be deformed to an ineffective toric orbifold.
The goal of this paper is to classify symplectic toric stratified spaces with isolated singularities. This extends a result of Burns, Guillemin, and Lerman which carries out this classification in the compact connected case. In making this classification, it is necessary to classify symplectic toric cones. Via a well-known equivalence between symplectic toric cones and contact toric manifolds, this allows for the classification of contact toric manifolds as well, extending Lerman's classification of compact connected contact toric manifolds.
In this paper we study whether symplectic toric manifolds are symplectically cohomologically rigid. Here we say that symplectic cohomological rigidity holds for some family of symplectic manifolds if the members of that family can be distinguished by their integral cohomology rings and the cohomology classes of their symplectic forms. We show how toric degenerations can be used to produce the symplectomorphisms necessary to answer this question. As a consequence we prove that symplectic cohomological rigidity holds for the family of symplectic Bott manifolds with rational symplectic form whose…
Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof i…
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains", the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all "concave toric domains", and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which …
According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $π$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and most of them are strongly symplectically fillable. The proof is based on the Lerman's classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of $T^d, d\geq3$ $(T…
We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture f…
A toric domain is a subset of $(\mathbb{C}^n,ω_{\text{std}})$ which is invariant under the standard rotation action of $\mathbb{T}^n$ on $\mathbb{C}^n$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free, and for which $c(U)=c(V)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the $\mathbb{T}^2$-action is free. Our theorem allows one to extend t…
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.
Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not tell whether the underlying symplectic manifolds are (non-equivariantly) symplectomorphic. For two dimensional torus actions on closed symplectic four-manifolds, we reduce the counting question to combinatorics, by expressing the manifold as a symplectic blowup in a way that is compatible with all the torus actions simultaneously. For this we use the theory o…
A striking result of McDuff and Schlenk asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional symplectic ball, the answer is governed by an "infinite staircase" determined by the odd-index Fibonacci numbers and the Golden Mean. Here we study embeddings of one four-dimensional symplectic ellipsoid into another, and we show that if the target is rational, then the infinite staircase phenomenon found by McDuff and Schlenk is quite rare. Specifically, in the rational case, there is an infinite staircase in precisely three ca…
We consider the embedding capacity functions $c_{H_b}(z)$ for symplectic embeddings of ellipsoids of eccentricity $z$ into the family of nontrivial rational Hirzebruch surfaces $H_b$ with symplectic form parametrized by $b\in [0,1)$. This function was known to have an infinite staircase in the monotone cases ($b= 0$ and $ b= 1/3$). It is also known that for each $b$ there is at most one value of $z$ that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked $b$-intervals, consisting of $b$-parameters where the embedding ca…
Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $ψ: W \to \frak{g}^*$, where $\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $ψ$ has fold singularities at points in the image of the folding hypersurface under the quotient map to $W$ and it is a unimodular local embedding away from these points. Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $ψ:W \to \frak{g}^*$, a unimodular map with folds. We …
We prove that every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces. A theorem of Abreu and Macarini shows that if the level set of the reduction passes through a non-displaceable set then the image of this set in the reduced space is also non-displaceable. Using this result we show that every symplectic toric orbifold contains a non-displaceable fiber and we identify this fiber.
We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains whic…
In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a $b$-symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition o…
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. We show that this result still holds in higher dimensions when we "stabilize" the embedding problem.
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.
A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map nee…
Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $Φ$. Assuming that $Φ$ is transverse to the ray through a given weight $\boldsymbolν$, associated to these data there is a complex $(d-r+1)$-dimensional polarized projective orbifold $\hat{M}_{\boldsymbolν}$ (referred to as the $\boldsymbolν$-th \textit{conic transform} of $M$). Namely, $\hat{M}_{\boldsymbolν}$ is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polariz…
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentum map, we obtain a complete classification of such manifolds in terms of decorated log affine polytopes, hence extending the classification of symplectic toric manifolds achieved by Atiyah, Guillemin-Sternberg, Kostant, and Delzant.
We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.
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 investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in $\mathbb R^4$. In particular, based on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are…
A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $S^1$-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capaci…
