arXiv 1301.45712013-01-23math.DGmath.SG

泊松几何中的Normal forms

Normal forms in Poisson geometry

Ioan Marcut

备注：: Utrecht University PhD thesis, 200 pages

摘要：

本文主要利用几何、形式和解析方法研究了辛叶周围Poisson结构的规范形。其中一个主要结果（定理2）是泊松几何中的标准形定理，它是局部Reeb稳定性（来自叶理理论）和切片定理（来自等变几何）的泊松几何版本。所得结果将Conn定理从不动点推广到任意辛叶.我们提出了两个证明这一结果：一个几何严重依赖于理论的李代数胚和李群胚（类似于新的证明康恩定理的Crainic和费尔南德斯），和分析一个使用纳什-莫泽快速收敛方法（更多的精神康恩的原始证明）。在解析方法的基础上，我们证明了紧Poisson子流形的局部刚性定理（定理4），这是Poisson几何中的第一个局部刚性定理。定理4在泊松结构的光滑变形研究中有一个令人惊讶的应用：在定理5中，我们计算Lie-Poisson球面（即对应于紧半单李代数的线性泊松流形内的不变单位球面）周围的泊松模空间。这是第一个这样的计算的泊松模空间的尺寸大于或等于3周围的退化（即非辛）泊松结构。本文的其他结果包括：辛实现存在性的一个新证明（定理0），辛叶理的一个规范形定理（定理1），Poisson子流形的一个形式规范形/刚性结果（定理3），以及李代数体上同调的驯服同伦算子的一般构造（驯服消失引理）。我们还重新康恩定理和汉密尔顿定理的刚性叶理。

英文摘要：

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. We present two proofs of this result: a geometric one relying heavily on the theory of Lie algebroids and Lie groupoids (similar to the new proof of Conn's theorem by Crainic and Fernandes), and an analytic one using the Nash-Moser fast convergence method (more in the spirit of Conn's original proof). The analytic approach gives much more, we prove a local rigidity result (Theorem 4) around compact Poisson submanifolds, which is the first of this kind in Poisson geometry. Theorem 4 has a surprising application to the study of smooth deformation of Poisson structures: in Theorem 5 we compute the Poisson-moduli space around the Lie-Poisson sphere (i.e. the invariant unit sphere inside the linear Poisson manifold corresponding to a compact semisimple Lie algebra). This is the first such computation of a Poisson moduli space in dimension greater or equal to 3 around a degenerate (i.e. non-symplectic) Poisson structure. Other results presented in the thesis are: a new proof to the existence of symplectic realizations (Theorem 0), a normal form theorem for symplectic foliations (Theorem 1), a formal normal form/rigidity result around Poisson submanifolds (Theorem 3), and a general construction of tame homotopy operators for Lie algebroid cohomology (the Tame Vanishing Lemma). We also revisit Conn's theorem and a theorem of Hamilton on rigidity of foliations.

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。

摘要：

英文摘要：

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