Higher jet evaluation transversality of $J$-holomorphic curves

Transversality and super-rigidity for multiply covered holomorphic curves

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically regular, and simple index zero curves in dimensions greater than four are generically super-rigid, implying e.g. that the Gromov-Witten invariants of Calabi-Yau 3-folds reduce to sums of local invariants for finite sets of embedded curves. We also establish partial results on super-rigidity in dimension four and regularity of branched covers, and briefly discu…

Generic transversality for unbranched covers of closed pseudoholomorphic curves

We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index zero. A corollary is that the Gromov-Witten invariants (without descendants) of symplectic 4-manifolds can always be computed as a signed and weighted count of honest J-holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homolo…

Punctured holomorphic curves and Lagrangian embeddings

We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant mod…

From Cascades to $J$-holomorphic Curves and Back

This paper develops the analysis needed to set up a Morse-Bott version of embedded contact homology (ECH) of a contact three-manifold in certain cases. In particular we establish a correspondence between "cascades" of holomorphic curves in the symplectization of a Morse-Bott contact form, and holomorphic curves in the symplectization of a nondegenerate perturbation of the contact form. The cascades we consider must be transversely cut out and rigid. We accomplish this by studying the adiabatic degeneration of $J$-holomorphic curves into cascades and establishing a gluing theorem. We note our g…

Moduli Spaces of $J$-holomorphic Curves with General Jet Constraints

In this paper, we prove that the tagent map of the holomorphic $k$- jet evaluation $j^k_{hol}$ from the mapping space to holomorphic $k$-jet bundle, when restricted on the universal moduli space of simple J-holomorphic curves with one marked point, is surjective. From this we derive that for generic $J$, the moduli space of simple $J$-holomorphic curves in class $β\in H_2(M)$ with general jet constraints at marked points is a smooth manifold of expected dimension.

Geometric transversality in higher genus Gromov-Witten theory

The construction of manifold structures and fundamental classes on the (compactified) moduli spaces appearing in Gromov-Witten theory is a long-standing problem. Up until recently, most successful approaches involved the imposition of topological constraints like semi-positivity on the underlying symplectic manifold to deal with this situation. One conceptually very appealing approach that removed most of these restrictions is the approach by K. Cieliebak and K. Mohnke via complex hypersurfaces, [CM07]. In contrast to other approaches using abstract perturbation theory, it has the advantage th…

The Gromov-Witten axioms for symplectic manifolds via polyfold theory

Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a well-defined Gromov-Witten invariant for $J$-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. The Gromov-Witten axioms, as originally described by Kontsevich and Manin, give algebraic relationships between the Gromov-Witten invariants. In this paper, we prove the Gromov-Witten axioms for the polyfold Gromov-Witten invariants.

$J$ -全纯曲线在一般$J$的Calabi-Yau流形中的嵌入性质

Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$

本文证明了：对于具有$n \geq 3$和它的第一个Chern类$c_1(M,ω) = 0$的辛流形$(M^{2n},ω)$上的驯服（或相容）殆复结构$J$的一般选择，从任何闭光滑Riemann曲面到$M$的所有某处内射$J$ -全纯映射都是\emph{embedded} .我们得到这个结果作为一般的最佳1-喷评价横截性结果的$J$ -全纯映射一般辛流形，我们也证明了在本文中。

Holomorphic jets in symplectic manifolds

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover, we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.

重审辛场论中的相容定向

Coherent orientations in symplectic field theory revisited

本文修正辛场论生成函数$\mathbf H$的符号，采用与Hutchings-Taubes一致的定向约定，从而在横截性假设下保证主方程$\mathbf H \cdot \mathbf H = 0$成立。

三维流形中嵌入伪全纯曲线的Compactness

Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds

我们证明了一个紧性定理的全纯曲线在四维辛，有嵌入投影的基础3-流形。它通过使用相交理论来加强SFT紧性定理的圆柱情况，以表明这种序列的退化永远不会产生多个覆盖或节点，因此很容易实现横截性。这适用于稳定的有限能量叶理理论，也提出了一种新的方法来定义接触3-流形，或更一般地说，3-流形稳定的哈密顿结构的SFT型不变量。