Positivity of Symplectic Area for Perturbed J-holomorphic Curves

Perturbed Holomorphic Cylinders, Exponential Estimates, and Flow Lines

A convergence result for perturbed holomorphic cylinders in $\mathbb{R}^{2n}$ is proved; a sequence of perturbed holomorphic cylinders converges to a configuration of two holomorphic disks joined by a flow line. A key step in the computation is played by an a priori exponential estimate.

Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II. Positivity, integrality and the gluing formula

We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gr…

卡拉比-丘流形上几何正性与负性的拓扑障碍

Topological obstructions to geometric positivity and negativity on Calabi-Yau manifolds

该研究证明卡拉比-丘流形在特定维数下不同胚于弱Fano簇，其基本群不同构于凯勒双曲流形的基本群，揭示了其与几何正性/负性类的拓扑刚性差异。

Intersection of almost complex submanifolds

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for $J$-holomorphic curves in almost complex $4$-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce $J$-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset o…

全纯盘与 GIT 商

Holomorphic disks and GIT quotients

本文研究全纯盘模空间与 GIT 商之间的对应关系，通过半稳定盘势公式计算商空间的盘势。

Contact 3-manifolds, holomorphic curves and intersection theory

This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed primarily at students and require only a minimal background in holomorphic curve theory, as the emphasis is on topological rather than analytical issues. Some of the gaps in the analysis are then filled in by the appendices, which include self-contained proofs of the similarity principle and positivity of intersections, and conclude with a "quick reference" fo…