Symmetric plane curves of degree 7 : pseudo-holomorphic and algebraic classifications

全纯量子模形式的代数方面

Algebraic aspects of holomorphic quantum modular forms

本文研究纽结体积猜想精细化中出现的矩阵值全纯量子模形式，提出其代数性质可由派生态积分的代数性质推导，并以$(-2,3,7)$-pretzel纽结为例进行了阐释。

Holomorphic curves in the 6-pseudosphere and cyclic surfaces

The space $\mathbf{H}^{4,2}$ of vectors of norm -1 in $\mathbb{R}^{4,3}$ has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form $G_2'$. In this paper we consider a class of holomorphic curves in $\mathbf{H}^{4,2}$ which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group are naturally parameterized by certain $G_2'$-Higgs bundles. Thi…

An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when $n$ is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into $\R^8$ equipped with an almost-complex structure that need not be…

Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the latter can be used to define pseudo-holomorphic submanifolds and subvarieties. The main theorem in this paper asserts that if the 4-manifold has a non zero Seiberg-Witten invariant, then the zero set of any given self-dual harmonic 2-form is the boundary of a pseudo-holomorphic sub…

Virtual neighborhood technique for pseudo-holomorphic spheres

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, ω, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \…

链级环路括号与伪全纯圆盘

Chain level loop bracket and pseudo-holomorphic disks

本文基于弦拓扑链模型与 Kuranishi 结构理论，详细论证了如何利用伪全纯圆盘模空间的虚拟基本链在链级构造环路括号的 Maurer-Cartan 元素，从而落实 Fukaya 在辛拓扑中的重要观察。