Toric Symplectic Manifolds Research Articles

Toric symplectic geometry and full spark frames

The collection of $d \times N$ complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces -- i.e., frames -- are used to give robust representations of complex-valued signals, so that geometrical and measure-theoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of $d$ of its columns gives a basis for $\mathbb{C}^d$? We show that the probability is one, generalizing recent work of Cahill, Mixon and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps -- certain spectral invariants of a frame -- and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature.

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

Loops in the fundamental group of which are not represented by circle actions

Abstract We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$ , where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$ , of symplectomorphisms that act trivially on homology, with $k \leq 4$ , is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $ , the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$ -spaces, and the computation of Seidel elements of some circle actions.

The Anosov–Katok method and pseudo-rotations in symplectic dynamics

We prove that toric symplectic manifolds admit Hamiltonian pseudo-rotations with a finite, and in a sense minimal, number of ergodic measures. The set of ergodic measures of these pseudo-rotations consists of the measure induced by the symplectic volume form and the Dirac measures supported at the fixed points of the torus action. Our construction relies on the conjugation method of Anosov and Katok.

Berezin–Toeplitz quantization in real polarizations with toric singularities

On a compact Kahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^+$.

ECH capacities, Ehrhart theory, and toric varieties

ECH capacities were developed by Hutchings to study embedding problems for symplectic $4$-manifolds with boundary. They have found especial success in the case of certain toric symplectic manifolds where many of the computations resemble calculations found in cohomology of $\mathbb{Q}$-line bundles on toric varieties, or in lattice point counts for rational polytopes. We formalise this observation in the case of convex toric lattice domains $X_\Omega$ by constructing a natural polarised toric variety $(Y_{\Sigma(\Omega)},D_\Omega)$ containing the all the information of the ECH capacities of $X_\Omega$ in purely algebro-geometric terms. Applying the Ehrhart theory of the polytopes involved in this construction gives some new results in the combinatorialisation and asymptotics of ECH capacities for convex toric domains.

The equivariant cohomology of complexity one spaces

Complexity one spaces are an important class of examples in symplectic geometry. They are less restrictive than toric symplectic manifolds. Delzant has established that toric symplectic manifolds are completely determined by their moment polytope. Danilov proved that the ordinary and equivariant cohomology rings are dictated by the combinatorics of this polytope. These results are not true for complexity one spaces. In this paper, we describe the equivariant cohomology for a Hamiltonian S^1\circlearrowright M^4 . We then assemble the equivariant cohomology of a complexity one space from the equivariant cohomology of the 2- and 4 -dimensional pieces, as a subring of the equivariant cohomology of its fixed points. We also show how to compute equivariant characteristic classes in dimension four.

Toric symplectic stacks

We give an intrinsic definition of compact 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 compact toric symplectic manifolds. As an application, we show that any compact toric symplectic stack can be deformed to an ineffective toric orbifold.

Toric actions in cosymplectic geometry

Abstract We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

Toric Contact Geometry in Arbitrary Codimension

Abstract We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Toric constructions of monotone Lagrangian submanifolds in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$

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.

Contact blow up and cylindrical contact homology of toric contact manifolds of Reeb type

Let (V,?) be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold (M,?). A contact blow up of (V,?) is the prequantization of a symplectic blow up of (M,?). Thus, a contact blow up of (V,?) is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blowup using only the cylindrical contact homology of the contact manifold we started with.

On the density function on moduli spaces of toric 4-manifolds

AbstractThe optimal density function assigns to each symplectic toric manifoldMa positive numberd ≤1 obtained by considering the ratio between the maximum volume ofMwhich can be filled by symplectically embedded disjoint balls and the total symplectic volume ofM. In the toric version,Mis toric and the balls need to be embedded respecting the toric action onM. We give a brief survey of toric symplectic manifolds and the recent constructions of moduli space structure on them, and then we recall how to define a natural density function on this moduli space. We review previous work which explains how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is 4.

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

Toric structures on bundles of projective spaces

Recently, extending work by Karshon et al., Borisov and McDuff showed in The topology of toric symplectic manifolds that a given closed symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, in The topology of toric symplectic manifolds McDuff also showed that a product of two projective spaces $\mathbb{C}P^r \times \mathbb{C}P^s$ with any given symplectic form has a unique toric structure provided that $r, s \geq 2$. In contrast, the product $\mathbb{C}P^r \times \mathbb{C}P^1$ can be given infinitely many distinct toric structures, although only a finite number of these are compatible with each given symplectic form $\omega$. In this paper, we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M) = 2$. In particular, all such manifolds are $\mathbb{C}P^r$ bundles over $\mathbb{C}P^s$ for some $r, s$. We show that there is a unique toric structure if $r \lt s$, and also that if $r, s \geq 2$, $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,\omega)$ has a unique toric structure, such as the case where $(M,\omega)$ is monotone.

Homological Mirror Symmetry and Fourier–Mukai Transform

We interpret symplectic geometry as certain theory by constructing a of curved A_\infty algebras which in some sense plays the role of a structure sheaf for symplectic manifolds. An interesting feature of this structure sheaf is that the symplectic form itself is part of its curvature term. Using this interpretation homological mirror symmetry can be understood by well-known duality theories in mathematics: Koszul duality or Fourier- Mukai transform. In this paper we perform the above constructions over a small open subset inside the smooth locus of a Lagrangian torus fibration. In a subsequent work we shall use the language of derived geometry to obtain a global theory over the whole smooth locus. However we do not know how to extend this construction to the singular locus. As an application of the local theory we prove a version of homological mirror symmetry between a toric symplectic manifold and its Landau-Ginzburg mirror.

Pseudotoric structures on toric symplectic manifolds

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds.

Remarks on Lagrangian intersections in toric manifolds

I will consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. The remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. I will show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits. This is joint work with Leonardo Macarini.

Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate.In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.

The topology of toric symplectic manifolds

This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff--Tolman concept of mass linear function. Using Timorin's description of the cohomology ring via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin's higher codimension barycenters.