Holomorphic triangles and invariants for smooth four-manifolds

Holomorphic triangle invariants and the topology of symplectic four-manifolds

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This …

Heegaard Floer 枕形中的全纯三角形

Holomorphic Triangles in the Heegaard Floer Pillowcase

本文研究Heegaard Floer枕形的全纯三角形，计算其标准有理斜率曲线三元组的三角形乘积与生成函数闭式，确定贡献类数量，并关联平面标记与双桥纽结的Floer群。

伪全纯三角形与中位数拟态

Pseudo-holomorphic triangles and the median quasi-state

本文证明全纯三角形最小面积受拉格朗日谱范数界定，并揭示其与中位数拟态的联系，进而给出泊松非交换性度量的尖锐上界及Calabi拟态亏缺下界。

Holomorphic polygons and smooth 4-manifold invariants

Any smooth, closed oriented 4-manifold has a surface diagram of arbitrarily high genus g>2 that specifies it up to diffeomorphism. The goal of this paper is to prove the following statement: For any smooth, closed oriented 4-manifold M, there is a sequence of weak A-infinity algebras indexed by g, and the homotopy equivalence class of each entry of this sequence is a diffeomorphism invariant of M.

Holomorphic curves in Exploded Torus Fibrations: Compactness

The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations corresponding to an algebraic family with normal crossing singularities are smooth.) In this paper we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov's compactness theorem in a topology strong enough for gluing analysis.

On Regularity of J-holomorphic Maps

We provide a short proof that an $L^2_1$ and $J$-holomorphic curve is in fact smooth. As an application, we deduce a removal of singularity theorem for curves of finite energy.