Universal moduli spaces in Gromov-Witten theory
Universal moduli spaces in Gromov-Witten theory
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格罗莫夫-维滕理论中的通用模空间
DOI：
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发表时间：
2013
期刊：
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通讯作者：
A. Gerstenberger
中科院分区：
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作者：
A. Gerstenberger
在Gromov-Witten理论中出现的（紧）模空间上构造流形结构和基本类是一个长期存在的问题。直到最近，大多数成功的方法都涉及在底层辛流形上施加拓扑约束（如半正性）来处理这种情况。一个概念上非常吸引人的方法
消除了大部分限制的方法是K。Cieliebak和K. Mohnke via complex hypersurfaces，[CM07].与其他使用抽象微扰理论的方法相比，它的优点是研究的对象仍然是定义在黎曼曲面上的全纯映射空间。
在这篇论文中，这种方法是推广的情况下，曲面的亏格0
在[CM 07]中处理一般情况。
在第一节中，引入了黎曼曲面空间，它取代了Deligne-Mumford空间，以处理后者是orbifolds的事实。此外，为了在后面的部分中使用，这些之间的相互关系
对于不同数量的标记点进行了澄清。在纤维化中截面的Sobolev空间的预备部分之后，在对Hamiltonian扰动和扰动曲线的相关模空间进行简短阐述之后，
构造一个将泛模空间分解为光滑Banach流形的方法。那里的重点主要在于整体方面的建设，因为当地的图片，即实际横截的普遍柯西-黎曼算子的零部分，是很好地理解。
然后，紧化这个模空间中存在的冒泡和后面的建设的动机，并给出了一个粗略的草图背后的基本思想。
在第一章的最后一部分，给出了必要的定义和结果，这些定义和结果是将文[CM 07]中关于具有切向条件的曲线的模空间的结果转移到文[CM 07]中所需要的。也有必要的限制几乎复杂的结构和哈密顿扰动从[IP 03]被纳入，后来允许使用的紧性定理证明在该参考。
在本文的最后一部分，这些结果然后被用来给出一个Gromov-Witten伪循环的定义，使用一个修改版本的模空间的曲线与额外的标记点，映射到一个复杂的超曲面从[CM 07]。然后证明这是定义良好的，使用
紧性定理从[IP 03]得到一个描述的边界和建设从前面的部分覆盖边界的流形的正确尺寸。
The construction of manifold structures and fundamental classes on the (compactifed) 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 that the objects to be studied still are spaces of holomorphic maps defined on Riemann surfaces.
In this thesis this approach is generalised from the case of surfaces of genus 0
dealt with in [CM07] to the general case.
In the first section the spaces of Riemann surfaces are introduced, that take the place of the Deligne-Mumford spaces in order to deal with the fact that the latter are orbifolds. Also, for use in the later parts, the interrelations of these
for different numbers of marked points are clarified. After a preparatory section on Sobolev spaces of sections in a fibration, the results presented there are then used, after a short exposition on Hamiltonian perturbations and the associated moduli spaces of perturbed curves, to
construct a decomposition of the universal moduli space into smooth Banach manifolds. The focus there lies mainly on the global aspects of the construction, since the local picture, i.e. the actual transversality of the universal Cauchy-Riemann operator to the zero section, is well understood.
Then the compactification of this moduli space in the presence of bubbling is presented and the later construction is motivated and a rough sketch of the basic idea behind it is given.
In the last part of the first chapter, the necessary definitions and results are given that are needed to transfer the results on moduli spaces of curves with tangency conditions from [CM07]. There also the necessary restrictions on the almost complex structures and Hamiltonian perturbations from [IP03] are incorporated, that later allow the use of the compactness theorem proved in that reference.
In the last part of this thesis, these results are then used to give a definition of a Gromov-Witten pseudocycle, using an adapted version of the moduli spaces of curves with additional marked points that are mapped to a complex hypersurface from [CM07]. Then a proof that this is well-defined is given, using the
compactness theorem from [IP03] to get a description of the boundary and the constructions from the previous parts to cover the boundary by manifolds of the correct dimensions.