Gluing pseudoholomorphic curves along branched covered cylinders I

Gluing pseudoholomorphic quilted disks

We construct families of quilted surfaces parametrized by the multiplihedra, and define moduli spaces of pseudoholomorphic quilted disks using the theory of pseudoholomorphic quilts of Wehrheim and Woodward. We prove a gluing theorem for regular, isolated pseudoholomorphic quilted disks. This analytical result is a fundamental ingredient for the construction of A-infinity functors associated to Lagrangian correspondences.

Gluing pseudoholomorphic curves along branched covered cylinders II

This paper and its prequel ("Part I") prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U_+ and U_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of U_+ at covers of gamma agrees with the total multiplicity of the positive ends of U_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U_+ and U_- to an index 2 curve by inserting genus zero branched co…

伪全纯曲线模空间的指数衰减估计与光滑性

Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves

本文针对带边界穿孔的黎曼面中的伪全纯曲线，证明了粘合解对颈区长度参数的导数具有指数衰减，并由此说明紧化模空间上 Kuranishi 结构的光滑性。

Enumerative vs. Symplectic Invariants and Obstruction Bundles

We give detailed descriptions of gluing pseudoholomorphic maps in symplectic geometry, especially in the presence of an obstruction bundle. The main motivation is to try to compare the symplectic and enumerative invariants of algebraic manifolds. These descriptions can also be used to enumerate rational curves with high-order degeneracies of local nature in projective spaces.

Pseudoholomorphic Curves Relative to a Normal Crossings Symplectic Divisor: Compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than…

编织曲面与小 exotic 4-流形

Braided Surfaces and Small Exotic 4-Manifolds

本文证明纤维化椭圆对合给出零手术结与环面乘积的二重分支覆盖，计算编织词，并推广至亏格二纤维化结，同时给出扩展障碍。