Questions in category: 辛几何 (Symplectic Geometry)

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## 1. $\Omega^2=\Omega_J^{+}\oplus\Omega_J^{-}$

Posted by haifeng on 2015-08-28 15:42:20 last update 2015-08-28 15:42:20 | Answers (0) | 收藏

$J:\ \alpha\mapsto\alpha^J,\quad \alpha^J(\cdot,\cdot):=\alpha(J\cdot,J\cdot).$

\begin{aligned} \Omega_J^{+}&:=\{\alpha\in\Omega^2(M)\mid\alpha^J=\alpha\},\\ \Omega_J^{-}&:=\{\alpha\in\Omega^2(M)\mid\alpha^J=-\alpha\}. \end{aligned}

## 2. [Thm]反对称双线性映射的标准形式

Posted by haifeng on 2013-07-04 23:05:20 last update 2013-07-05 13:18:48 | Answers (2) | 收藏

$u_1,\ldots,u_k; e_1,\ldots,e_n; f_1,\ldots,f_n;$

$\begin{array}{ll} \Omega(u_i,v)=0,& \forall\ i, \forall\ v\in V\\ \Omega(e_i,e_j)=0=\Omega(f_i,f_j),& \forall\ i,j\\ \Omega(e_i,f_j)=\delta_{ij},& \forall\ i,j \end{array}$

$\Omega(u,v)=[--- u ---] \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & Id\\ 0 & -Id & 0 \end{pmatrix} \begin{bmatrix} |\\ v\\ | \end{bmatrix}$

$\begin{array}{rcl} \widetilde{\Omega}:\ V&\rightarrow& V^*\\ v&\mapsto&\widetilde{\Omega}(v) \end{array}$

Def. 如果 $\widetilde{\Omega}$ 是双射, 即 $U=\{0\}$, 则称反对称双线性形式 $\Omega$ 是辛的(symplectic)或是非退化的(nondegenerate). 此时称 $\Omega$ 为向量空间 $V$ 上的线性辛结构. $(V,\Omega)$ 称为辛向量空间.

## 3. 辛几何参考书

Posted by haifeng on 2013-05-18 09:46:56 last update 2014-04-12 12:40:15 | Answers (0) | 收藏

Michele Audin , Jacques Lafontaine ,

## Michèle Audin

http://www-irma.u-strasbg.fr/~maudin/

Holomorphic curves in symplectic geometry (Progress in Mathematics)

http://www.amazon.com/Holomorphic-Symplectic-Geometry-Progress-Mathematics/dp/3764329971

This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov\'s compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.

## 4. [Def]辛流形 $(M,\omega)$ 上的近复结构称为 $\omega$-calibrated 的.

Posted by haifeng on 2012-08-06 10:48:35 last update 2017-02-28 18:35:48 | Answers (0) | 收藏

$\omega(JX,JY)=\omega(X,Y),\quad\omega(JV,V) > 0,$

$\omega[x](v,w)=\omega[x](J_x v, J_x w),\quad\omega[x](u,J_x u) > 0,$

Remark:

$M$ 上与 $\omega$ 相适应的近复结构 $J$ 的存在性是一个经典的结论（参见[2], Lecture 2）

References:

[1] P. Delanoë, Sur L'analogue presque-complexe de l'equation de Calabi-Yau. Osaka J. Math. 33 (1996), 829-846.

[2] A. Weinstein, Lectures on symplectic manifolds, American Math. Society 1977 (CBMS regional conference series in math. #29).

## 5. 辛流形上被辛形式 $\omega$ 驯化的所有近复结构构成的集合是可缩的.

Posted by haifeng on 2012-08-06 10:24:42 last update 2012-08-09 15:20:49 | Answers (2) | 收藏

$\begin{array}{rcl} \Phi:\ \mathfrak{J}_t(\omega)&\rightarrow&B=\{S\in\mathcal{M}_{n\times n}(\mathbb{R})\mid J_0 S+S J_0=0,\ \text{且}\ \|S\| < 1\}\\ J&\mapsto& (J+J_0)^{-1}\circ(J-J_0) \end{array}$

$\begin{array}{rcl} \Phi:\ \mathfrak{J}_c(\omega)&\rightarrow&B=\{S\in\mathcal{M}_{n\times n}(\mathbb{R})\mid J_0 S+S J_0=0,\ S^T=S,\ \text{且}\ \|S\| < 1\}\\ J&\mapsto& (J+J_0)^{-1}\circ(J-J_0) \end{array}$

Remark:

$\omega[x](v,J_x v) > 0$

## 6. 设 $(M,\omega)$ 是 $2n$ 维闭的辛流形, $\omega$ 是辛形式, 从而代表了 $H_{\mathrm{dR}}^2(M)$ 中的元素 $[\omega]$. 证明 $[\omega]\neq 0$.

Posted by haifeng on 2012-08-06 00:19:03 last update 2012-08-06 00:21:16 | Answers (1) | 收藏

Hint. 这可从问题900 直接推出.

## 7. 设 R 是单变量解析函数的黎曼曲面, 刻画 R 的万有覆盖.

Posted by haifeng on 2012-03-15 16:49:28 last update 2012-03-15 16:49:28 | Answers (0) | 收藏

E 到自身的所有共形映射构成一个群 $\Omega$. 而 R 的基本群可由 $\Omega$ 的一个子群 $\Delta$ 忠实地表示, 这个子群 $\Delta$ 在 E 上是不连续的.

$f(z_1)=z_1,\quad f^{-1}(z_1)=z_1.$

References:

Carl Ludwig Siegel, Symplectic Geometry. Institute for advanced study, Princeton, N. J.

## 8. 自守函数理论推广至多变量情形

Posted by haifeng on 2012-03-15 16:48:39 last update 2018-10-22 17:03:14 | Answers (0) | 收藏

1) 找出 $m$ 维复向量空间中所有有界的 simple domain E. 这个 E 是对应于某个多变元解析映射的, 其所对应的共形映射全体构成群 $\Omega$, 关于这个群, E 是对称空间.

2) 研究 E 的不变几何性质,  找出 $\Omega$ 中的不连续子群 $\Delta$, 并构建它们的基本域(fundamental domains).

3) To study the field of automorphic functions in E with the group $\Delta$.

References:

Carl Ludwig Siegel, Symplectic Geometry. Institute for advanced study, Princeton, N. J.

## 9. Symplectic cone

Posted by haifeng on 2012-01-06 21:06:54 last update 2012-01-06 21:07:46 | Answers (0) | 收藏

$cc:\ \Omega_M\longrightarrow H^2(M;\mathbb{R})$

References:

T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom., 17 (2009), 651--684.

## 10. 找出不具有 Kähler 结构的紧致辛流形.

Posted by haifeng on 2012-01-06 18:57:07 last update 2012-01-06 18:57:07 | Answers (0) | 收藏

Thurston 给出了一个 4 维的例子.

References:

A. Weinstein, Lectures on symplectic manifolds, C.B.M.S. regional conference series, 29, A.M.S. Rhode Island, 1977.

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