arXiv 1601.040482016-01-15math.SG

Genus zero Gromov-Witten axioms via Kuranishi atlases

Robert Castellano

备注：: 38 pages

英文摘要：

A Kuranishi atlas is a structure used to build a virtual fundamental class on moduli spaces of $J$-holomorphic curves. They were introduced by McDuff and Wehrheim to resolve some of the challenges in this field. This paper completes the construction of genus zero Gromov-Witten invariants using Kuranishi atlases and proves the Gromov-Witten axioms of Kontsevich and Manin. To do so, we introduce the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The topology of Kuranishi atlases

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Starting from the same core idea (patching local finite dimensional reductions) we develop a theory of topological Kuranishi atlases and cobordisms that transparently resolves algebraic and topological challenges in this virtual regularization approach. It applies to any Kuranishi-type setting, e.g. atlases with isotropy, boundary and corners, or lack of differentiable structure.

2015-08-11math.SG

Smooth Kuranishi atlases with isotropy

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions, given by smooth sections that are equivariant under a finite isotropy group. Building on our notions of topological Kuranishi atlases and perturbation constructions in the case of trivial isotropy, we develop a theory of Kuranishi atlases and cobordisms that transparently resolves the challenges posed…

2015-08-11math.SG

Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces

Kuranishi atlases were introduced by McDuff and Wehrheim to build a virtual fundamental class on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This paper considers Gromov-Witten moduli spaces and shows they admit a smooth enough Kuranishi atlas to be able to define a Gromov-Witten virtual fundamental class in any virtual dimension. The key step for this result is the proof of a stronger gluing theorem.

2016-05-01math.SG

The fundamental class of smooth Kuranishi atlases with trivial isotropy

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program. We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we …

2015-08-11math.SG

Kuranishi atlases with trivial isotropy - the 2013 state of affairs

Kuranishi structures were introduced to symplectic topology by Fukaya and Ono and recently refined by Joyce, in order to extract homological data from compactified moduli spaces of holomorphic maps in cases where geometric regularization approaches such as perturbations of the almost complex structure do not yield a smooth structure on the moduli space. We give a general survey of regularization techniques in symplectic topology, pointing to some general analytic issues, and discussing some specific topological issues of the Kuranishi approach. In the main body of the paper we provide an abstr…

2015-08-11math.SG

Notes on Kuranishi Atlases

These notes aim to explain a joint project with Katrin Wehrheim that uses finite dimensional reductions to construct a virtual fundamental class for the Gromov--Witten moduli space of closed genus zero curves. Our method is based on work by Fukaya and Ono as well as more recent work by Fukaya, Oh, Ohta, and Ono. We reformulated their ideas in order to clarify the formal structures underlying the construction and make explicit all important choices (of tamings, shrinkings and reductions), thus creating tools with which to give an explicit proof that the virtual fundamental class is independent …

2016-09-25math.SG

更多相关论文收起更多论文

Gromov-Witten不变量之间的等价性：域依赖扰动与Kuranishi结构方法

Equivalence between Gromov-Witten invariants: Domain dependent perturbation and Kuranishi structure approaches

本文证明了在Gromov-Witten不变量理论中，域依赖扰动构造的有理伪循环与Kuranishi结构定义的虚拟基本类诱导出相同的同调类，从而建立了两种方法的等价性。

2025-07-07math.SG

通过全局Kuranishi图表定义的Gromov-Witten不变量的性质

Properties of Gromov-Witten invariants defined via global Kuranishi charts

本文通过全局Kuranishi图表定义了通用辛流形的引力后裔和equivariant Gromov-Witten不变量，并证明其满足Kontsevich和Manin的公理，同时推导了Hamiltonian流形的显式公式。

2026-03-03math.SG

辛Gromov-Witten理论中的全局Kuranishi图表与乘积公式

Global Kuranishi charts and a product formula in symplectic Gromov-Witten theory

本文为闭辛流形上所有亏格的稳定伪全纯映射模空间构造了全局Kuranishi图表，借此证明了辛Gromov-Witten不变量的乘积公式，并得出量子上同调的Künneth公式。

2024-07-24math.AGmath.SG

Gromov-Witten理论中Kuranishi结构的退化和胶合以及开Gromov-Witten不变量在辛割下的退化/胶合公理

Degeneration and gluing of Kuranishi structures in Gromov-Witten theory and the degeneration/gluing axioms for open Gromov-Witten invariants under a symplectic cut

本文研究了Gromov-Witten理论中Kuranishi结构在辛割下的退化和胶合。这导致我们的退化公理和粘合公理开放Gromov-Witten不变量。然后，它们提供了一条通过专业化构建虚拟基础链的途径。附录中给出了Li-Ruan/Li与Ionel-Parker关于封闭Gromov-Witten不变量的退化公式的等价性的评论。

2006-09-18hep-thmath.AGmath.SG

Gromov-Witten theory via Kuranishi structures

In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $ψ$-classes and Hodge classes.

2017-01-26math.AGmath.SG

Strict orbifold atlases and weighted branched manifolds

This note revisits the ideas in an earlier (2007) paper on orbifolds and branched manifolds, showing how the constructions can be simplified by using a version of the Kuranishi atlases recently developed by McDuff--Wehrheim. We first show that every orbifold has such an atlas, and then use it to obtain explicit models first for the nonsingular resolution of an oriented orbifold (which is a weighted nonsingular groupoid with the same fundamental class) and second for the Euler class of an oriented orbibundle. In this approach, instead of appearing as the zero set of a multivalued section, the E…

2015-11-15math.SG

$S^1$ -等变对上具有Lagrange边界条件和开Gromov-Witten不变量的J-全纯曲线的模

Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

设$(X,ω)$是辛流形，$J$是$ω$ -驯服的殆复结构，$L$是拉格朗日子流形. $J$ -全纯曲线在$L$中的模空间的稳定紧化是紧的，并且是Gromov $C^\infty$ -拓扑中的Hausdorff.我们构建了一个Kuranishi结构与角的意义上的谷和小野。如果$L$是自旋的，那么这个Kuranishi结构是可定向的。在模空间的期望维数为零的特殊情况下，有一个$S^1$行动的对$(X,L)$，保持$J$和自由的$L$，我们定义的欧拉数为这个$S^1$等变对和规定的拓扑数据。我们推测，这个有理数是一个计算本地化技术使用给定的$S^1$行动。

2004-12-04math.SG

Holomorphic curves in exploded manifolds: virtual fundamental class

We define Gromov--Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class $[\mathcal K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general version of an embedded Kuranishi structure.) We also show how to integrate differential forms over $[\mathcal K]$ to obtain numerical invariants, and push forward differential forms from $\mathcal K$ over suitable evaluation maps. We show that such invariants are independent of any choices, and are compatible with pullbacks, products, and tropical completion of Kur…

2018-10-16math.DGmath.SG

减少的Gromov-Witten不变量无需鬼气气泡限制

Reduced Gromov-Witten invariants without ghost bubble censorship

Rutgers University（罗格斯大学）

本文通过有效支持的多值扰动定义了所有亏格的减少Gromov-Witten不变量，绕过了Zinger和Ekholm-Shende的硬分析结果。

2026-07-16math.AGmath.SG

复与Morava局部$K$-理论中的Gromov-Witten不变量

Gromov-Witten invariants in complex and Morava-local $K$-theories

本文为闭辛流形构造取值于复$K$-理论及Morava局部上同调理论的Gromov-Witten型不变量，证明其满足Kontsevich-Manin公理，并借助全局Kuranishi图和计数理论定义量子$K$-理论作为上同调环的交换形变。

2024-07-16math.AGmath.SG

与 Lagrangian 子流形相关的开闭 Deligne–Mumford 场论

An open-closed Deligne-Mumford field theory associated to a Lagrangian submanifold

本文构造任意亏格、边界在 Lagrangian 子流形上的全纯映射全局 Kuranishi 图，并据此建立统一 Fukaya 代数与 Gromov–Witten 理论的开闭 Deligne–Mumford 场论。

2025-01-08math.SG

轨形上的一个新横截性条件与整数值Gromov-Witten型不变量

A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants

本研究引入轨形向量丛截面的FOP横截性条件，可定义伪全纯曲线模空间的整虚闭链，为紧致辛流形整数值Gromov-Witten型不变量定义及哈密顿纤维化上同调分裂定理证明提供新工具。

2025-02-21math.AGmath.ATmath.SG