Smooth Kuranishi structures with trivial isotropy
Smooth Kuranishi structures with trivial isotropy
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具有微不足道各向同性的平滑 Kuranishi 结构
DOI：
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发表时间：
2012
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通讯作者：
K. Wehrheim
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作者：
D. Mcduff;K. Wehrheim
为任何Kuranishi结构构建虚拟基础类，将在以下部分中概述。虽然[HWZ 1，HWZ 2，HWZ 3]中的多重折叠语言和正则化定理的证明是高度涉及的，但它似乎是详细开发的，并且很容易引用。基本哲学和语言的调查现在可以在[GFFW]中找到。正如在下面第4节中讨论的给定全纯曲线模空间的Kuranishi结构的构造一样，多重正则化方法的应用仍然需要将紧化模空间描述为多重丛中Fredholm截面的零集。正是在这里，多重方法有望在正则化技术中取得最具革命性的进步。首先，模空间的纤维积在多重刻划下被自然地刻划为多重刻划的乘积上的Fredholm截面的零集。例如，可以通过将SFT的多重设置与莫尔斯轨迹空间上的平滑结构相结合来获得PSS态射的多重设置，参见[AFW]。其次，Hofer-Wysocki-Zehnder目前正致力于形式化一个“模块化”的方法，以这样一种方式，分析设置可以在本地域和目标的每一个奇异类型。这样，以不同的方式组合先前处理过的奇点的新模空间的多重折叠设置只需要域和目标的底层空间的Deligne-Mumford型理论。注2.3.1.虽然多重框架是从全纯曲线模空间构造代数不变量的一个非常强大的方法，但它在几何应用中也有一些缺陷。·对于涉及正则化后解的几何性质的参数，需要谨慎对待。其原因是扰动解不是解偏微分方程，而是解Cauchy-Riemann方程的抽象紧扰动。本质上，我们只能处理这样一个事实，即扰动解可以在与标度拓扑相容的任何度量（例如，在闭曲线的情况下，任何Ck度量）中任意接近未扰动解。·尽管重参数化在映射空间上尺度平滑地起作用，但光滑、自由、适当作用的等变正则化问题由于与收缩的相互作用（即胶合构造）而仍然是非平凡的。在自治哈密尔顿算子的Floer轨迹空间上的S1作用的例子中，虚维度为0的未正则化紧化Floer轨迹空间可能包含断裂轨迹。商空间的相应层，
construction of a virtual fundamental class for any Kuranishi structure that will be outlined in the following sections. While the language of polyfolds and the proof of the regularization theorems in [HWZ1, HWZ2, HWZ3] is highly involved, it seems to be developed in full detail and is readily quotable. A survey of the basic philosophy and language is now available in [GFFW]. Just as in the construction of a Kuranishi structure for a given holomorphic curve moduli space discussed in Section 4 below, the application of the polyfold regularization approach still requires a description of the compactified moduli space as the zero set of a Fredholm section in a polyfold bundle. It is here that the polyfold approach promises the most revolutionary advance in regularization techniques. Firstly, fiber products of moduli spaces with polyfold descriptions are naturally described as zero sets of a Fredholm section over a product of polyfolds. For example, one can obtain a polyfold setup for the PSS morphism by combining the polyfold setup for SFT with a smooth structure on Morse trajectory spaces, see [AFW]. Secondly, Hofer–Wysocki–Zehnder are currently working on formalizing a “modular” approach to the polyfold axioms in such a way that the analytic setup can be given locally in domain and target for every singularity type. With that, the polyfold setup for a new moduli space that combines previously treated singularities in a different way would merely require a Deligne–Mumford type theory for the underlying spaces of domains and targets. Remark 2.3.1. While the polyfold framework is a very powerful method for constructing algebraic invariants from holomorphic curve moduli spaces, it also has some pitfalls in geometric applications. • Some caution is required with arguments involving the geometric properties of solutions after regularization. The reason for this is that the perturbed solutions do not solve a PDE but an abstract compact perturbation of the Cauchy–Riemann equation. Essentially, one can only work with the fact that the perturbed solutions can be made to lie arbitrarily close to the unperturbed solutions in any metric that is compatible with the scale-topology (e.g. any Ck-metric in the case of closed curves). • Despite reparametrizations acting scale-smoothly on spaces of maps, the question of equivariant regularization for smooth, free, proper actions remains nontrivial due to the interaction with retractions, i.e. gluing constructions. In the example of the S1-action on spaces of Floer trajectories for an autonomous Hamiltonian, the unregularized compactified Floer trajectory spaces of virtual dimension 0 may contain broken trajectories. The corresponding stratum of the quotient space,