Residues of the Bott class and an application to the Futaki invariant
Residues of the Bott class and an application to the Futaki invariant
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Bott 类的残基和 Futaki 不变量的应用
DOI：
10.4310/ajm.2003.v7.n2.a6
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发表时间：
2003
影响因子：
0.6
通讯作者：
T. Asuke
中科院分区：
文献类型：
--
作者：
T. Asuke
讨论了横向全纯叶化的 Bott 类和复流形的 Futaki 不变量。主要工具是 Heitsch 残差的推广和 Lehmann 引入的 Cech-de Rham 协链的集成。结果表明，Bott 类承认自然分解是 Cheeger 和 Simons 意义上的微分特征。分解意味着从动力系统的角度来看，Futaki 不变量的实部和复部具有一定的差异。介绍。在特征类理论中，所谓的 Chern-Simons 类非常重要。在本文中，研究了 Futaki 不变量，更一般地说，研究了横向全纯叶状结构的 Bott 类。 Bott 类是横向全纯叶理最基本的特征类。如果叶状结构的全同态是用双全纯微分同胚来写的，那么这里的叶状结构就被称为横向全纯。 Bott 类是叶状结构复杂法向束的 Chern-Simons 类或 CheegerSimons 类。在对复法向丛的附加假设下，Bott 类的定义变得与 Godbillon-Vey 类的定义完全平行。从这个意义上说，Bott 级是复杂类别中 Godbillon-Vey 级的对应物。然而，它的属性尚未得到很好的理解。困难之一是 Bott 类是 C/Z 系数上同调类的元素。事实上，虚部可以用封闭的形式来写，并且更容易研究，而实部的研究通常需要一些额外的技术或概念，例如77-不变量。在本文中，我们首先介绍 Cech-de Rham 复形的修改版本，并证明 Bott 类的某个代表可以被视为微分形式（定义 2.13 和定理 2.19）。这是基于 [23] 中发现的关于 Cech-de Rham 复合体的 Chern-Weil 理论。已经有一些方法来考虑除 C 或 R 之外的系数的 de Rham 上同调 [11]、[21]，其中使用单纯复形。根据我们的表述，Lehmann [19] 引入的 Cech-de Rham 共链的集成发挥着重要作用，它使构建变得简单。主要结果之一是对 Bott 类的 Heitsch 残差的推广（定理 3.4）。结合 Ghys-Gomez-Mont-Saludes 的复维一叶状结构定理，该残差给出了 Dummy 定理的弱形式推广（推论 5.4）。作为留数在更高维叶状结构中的应用，我们讨论了 Futaki 不变量。它是在复流形的双全纯自同构群上定义的群特征，并以 C/Z 计值。给定复流形 M 的自同构 a，我们可以自然地构造一个称为悬浮的叶流形 M^，而 Futaki 不变量本质上是该叶流形的 Bott 类。 Futaki不变量的计算经常涉及77不变量，这需要一些额外的结构。计算结果 * 2003 年 3 月 10 日收到； 2003 年 5 月 30 日接受出版。 + 京都大学数学系（吉田南分部），京都大学，京都 606-8501，日本 (asuke@math.kyoto-u.ac.jp)。文部科学省资助，批准号：13740042、15740036。
The Bott class for transversely holomorphic foliations and the Futaki invariant for complex manifolds are discussed. The main tools are a generalization of Heitsch's residue and the integration of Cech-de Rham cochains introduced by Lehmann. It is shown that the Bott class admits a natural decomposition as a differential character in the sense of Cheeger and Simons. The decomposition implies that the real and the complex parts of the Futaki invariant have a certain difference from the viewpoint of dynamical systems. Introduction. In the theory of characteristic classes, so-called Chern-Simons classes are very important. In this paper, the Futaki invariant and more generally, the Bott class of transversely holomorphic foliations are studied. The Bott class is the most fundamental characteristic class for transversely holomorphic foliations. Here a foliation is said to be transversely holomorphic if its holonomy is written in terms of biholomorphic diffeomorphisms. The Bott class is a Chern-Simons class or CheegerSimons class for the complex normal bundle of the foliation. Under an additional assumption on the complex normal bundle, the definition of the Bott class becomes completely parallel to that of the Godbillon-Vey class. In this sense, the Bott class is a counterpart of the Godbillon-Vey class in the complex category. However, its properties are not yet well-understood. One of difficulties is that the Bott class is an element of C/Z-coefficient cohomology class. Indeed, the imaginary part can be written by a closed form and is easier to study, while the study of the real part often needs some additional techniques or notions, e.g. 77-invariants. In this paper, first we introduce a modified version of Cech-de Rham complex and show that a certain representative of the Bott class can be treated as a differential form (Definition 2.13 and Theorem 2.19). This is based on the Chern-Weil theory on the Cech-de Rham complex found in [23]. There have been already some approaches to consider the de Rham cohomology with coefficients other than C or R [11], [21], where simplicial complexes are used. Under our formulation, the integration of Cech-de Rham cochains introduced by Lehmann [19] plays an essential role and it allows the construction to be elementary. One of the main results is a generalization of Heitsch's residue of the Bott class (Theorem 3.4). Combined with a structure theorem of complex codimension-one foliations due to Ghys-Gomez-Mont-Saludes, this residue gives a generalization of Dummy's theorem in a weak form (Corollary 5.4). As an application of the residue to higher codimensional foliations, the Futaki invariant is discussed. It is a group character defined on the groups of biholomorphic automorphisms of a complex manifold and is valued in C/Z. Given an automorphism a of a complex manifold M, one can naturally construct a foliated manifold M^ called the suspension, and the Futaki invariant is essentially the Bott class of this foliation. Calculations of the Futaki invariant often involve the 77-invariant, which need some additional structures. The calculations * Received March 10, 2003; accepted for publication May 30, 2003. + Department of Mathematics (Yoshida-Minami branch), Kyoto University, Kyoto 606-8501, Japan (asuke@math.kyoto-u.ac.jp). Supported by Ministry of Education, Culture, Sports, Science and Technology, Grant No. 13740042, 15740036.