Remarks on Lagrangian intersections in toric manifolds

Minimal and Hamiltonian-minimal submanifolds in toric geometry

In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real quadrics. In particular, we explain the nature of this phenomenon by proving H-minimality in a more conceptual way, and prove minimality of the same submanifolds in the corresponding moment-angle manifolds.

Periodic leaf-wise intersection points from Lagrangians

We investigate leaf-wise intersection points on hypersurfaces of contact type in monotone symplectic manifolds. We show that monotone Floer-essential Lagrangians detect periodic leaf-wise intersection points in hypersurfaces of contact type whose Reeb flow is Zoll. Examples include the prequantization bundles appearing in monotone toric negative line bundles. Generalizing, we prove the existence of leaf-wise intersection points for certain annulus subbundles in weak+-monotone negative line bundles, not necessarily toric. The proofs combine reduced symplectic cohomology with the original method…

非常仿射完全交的拉格朗日骨架

Lagrangian skeleta of very affine complete intersections

研究$(\mathbb{C}^*)^n$内完全交$Z^\circ$的拉格朗日骨架，通过计算其骨架并描述分解，应用结果在大体积极限下为巴蒂列夫 - 鲍里索夫对建立同调镜像对称，还证明等价性与环面HMS兼容及相关函子性质。

On symplectic capacities of toric domains

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…

四维凸环面域辛嵌入问题中的障碍

Obstructions in Symplectic Embedding Problems for Four-Dimensional Convex Toric Domains

本文推广Hutchings的ECH方法，研究四维凸环面域及其并集的辛嵌入障碍，并利用交正性发现新障碍，阻碍嵌入间同伦的存在。

Non-displaceable toric fibers on compact toric manifolds via tropicalizations

We give a combinatorial way to locate non-displaceable Lagrangian toric fibers on compact toric manifolds. By taking the intersection of certain tropicalizations coming from combinatorial data of a moment polytope, we locate all strongly bulk-balanced fibers introduced in [FOOOSurv]. As an application, we show that every bulk-balanced fiber defined in [FOOOToric2] is strongly bulk-balanced. Thus, the method indeed detects the positions of all non-displaceable fibers that can be detected by Lagrangian Floer theory developed in [FOOOToric1] and [FOOOToric2].