Tropical Lagrangian Hypersurfaces are Unobstructed

热带几何中的可实现性与拉格朗日子流形的无障碍性

Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

本文提出热带子簇的 $B$-可实现性概念，证明若拉格朗日子流形提升具有定义良好的 Floer 上同调且可通过界定上链变为无障碍，则该热带子簇是 $B$-可实现的，并应用于证明亏格零热带曲线及热带阿贝尔曲面内热带曲线的可实现性。

三维热带曲线的刚性与可实现性

Rigidity and Realizability for Tropical Curves in Dimension 3

本研究利用伪全纯圆盘边界的$H_1(L)$类提出拉格朗日三维流形的无障碍判据，证明具裤对分解的刚性热带曲线的拉格朗日提升无障碍，并在同调镜像对称成立的前提下推出镜像阿贝尔三维流形中存在对应实现。

Lagrangian submanifolds from tropical hypersurfaces

We prove that a smooth tropical hypersurface in $\mathbb{R}^3$ can be lifted to a smooth embedded Lagrangian submanifold in $(\mathbb{C}^*)^3$. This completes the proof of the result announced in the article "Lagrangian pairs pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of pants as the main building blocks.

Calabi-Yau components in general type hypersurfaces

For a one-parameter family of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers using tropical geometry. We show that after normalization, each holomorphic n-form is approximately supported on a unique open component and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our re…

Tropical Lagrangians in toric del-Pezzo surfaces

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $(\mathbb{C}^*)^n$ whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $L_{T^2}$ in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $(\mathbb{CP}^2\setminus E, W), \c…

多维热带级数上的收缩动力学

Shrinking dynamic on multidimensional tropical series

本文研究紧凸域上热带级数的收缩动力学，证明单点收缩算子的公平顺序有限步稳定，并给出事件驱动实现及近似路径的奇点性质。