Jean-Claude Hausmann, Susan Tolman
We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.
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Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations $\mathrm{Gr}_2(\C^n)^{\tree}$ of $\mathrm{Gr}_2(\C^n)$ to each trivalent tree $\tree$ with $n$ leaves. These degenerations induce toric degenerations $M_{\br}^{\tree}$ of $M_{\br}$, the space of $n$ ordered, weighted (by $\br$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons." We prove the conjecture of Foth and Hu [FH] that the toric f…
We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.
We count the conjugacy classes of maximal tori in the groups of symplectomorphisms of S^2 \times S^2 and of the blow-up of CP^2 at a point.
I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-contact type one obtains a configuration of Sasaki cones called a bouquet such that each Sasaki cone is associated with a conjugacy class of tori of Reeb type.
In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over Hirzebruch surfaces. Note that the contact structures in question are fillable. This may be contrasted with an earlier paper where we showed that there are infinitely many non-conjugate tori in the contactomorphism groups of certain overtwisted lens spaces (Contact cuts, Israel …
By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its preimage under the extension map is a maximal torus not only in the group $\Diff(P, α)$ of diffeomorphisms of $P$ preserving $α$ but also in the much bigger group of contactomorphisms $\Diff (P, ξ)$, the group of diffeomorphism of $P$ preserving the contact distribution $ξ= \ker α$…
This paper studies the local and global aspects of semi-toric integrable systems, introduced by Vu Ngoc, using ideas stemming from the theory of Hamiltonian S^1-spaces developed by Karshon. First, we show how any labeled convex polygon associated to a semi-toric system (as defined by Vu Ngoc) determines Karshon's labeled directed graph which classifies the underlying Hamiltonian S^1-space up to isomorphism. Then we characterize adaptable semi-toric systems, i.e. those whose underlying Hamiltonian S^1-action can be extended to an effective Hamiltonian T^2-action, as those which have at least on…
We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that…
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.
We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb C^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group U(n). We prove that the first Chern classes of the $n$ line bundles associated with the fibration $r$-level set $\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \mathbb C).$ By applying the Duistermaat--Heckman Theorem, we then deduce the relat…
Disk potential functions for polygon spaces
针对等边且一般的多边形空间，推导其Floer理论SYZ镜像，通过计算非平凡开Gromov-Witten不变量得到 caterpillar 弯曲系统的单调环面纤维的圆盘势函数，并结合已有工作获得弯曲系统的圆盘势函数，构造无冻结变量的A型镜像簇。
We determine the integral homology of the orbit space of a maximal compact torus action on the Grassmannian Gr(2,5). Our approach uses the well-known Geometric Invariant Theory of the maximal algebraic torus action on this Grassmannian.
Fibrations by Lagrangian tori for maximal Calabi-Yau degenerations and beyond
介绍在凯勒流形退化中构造拉格朗日环面纤维化的通用技术，该技术利用阿坎波空间边界及辛联络，证明极大卡拉比 - 丘退化附近纤维一般区域有符合期望渐近性质的拉格朗日环面纤维化。
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