On normal forms of complex points of small C²-perturbations of real 4-manifolds embedded in a complex 3-manifold

Figures & data

Table 1. Orbits of the action (2(2) $\mathrm{\Xi }:\left((c,P),(A,B)\right)\mapsto (c{P}^{\ast }AP,\overline{c}{P}^{T}BP),P\in G{L}_n\left(\mathbb{C}\right),c\in {\mathbb{C}}^{\ast }.$(2) ) for n = 2.

$\dim$	A	B	A	B	A	B	A	B
9	$1\oplus e^{i\theta }$	$\left[\begin{array}{cc}0& b\\ b& d\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ \tau & 0\end{array}\right]$	$\left[\begin{array}{cc}{e}^{i\varphi }& b\\ b& \zeta \end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& i\end{array}\right]$	$\left[\begin{array}{cc}0& b\\ b& 0\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$	$a\oplus 1$
		$\left[\begin{array}{cc}a& r{e}^{i\varphi }\\ r{e}^{i\varphi }& d\end{array}\right]$		$\left[\begin{array}{cc}0& b\\ b& e^{i\varphi }\end{array}\right]$		$a\oplus \zeta$		$\left[\begin{array}{cc}\zeta & b\\ b& 1\end{array}\right]$
		$\left[\begin{array}{cc}a& b\\ b& 0\end{array}\right]$		$1\oplus \zeta$				$\left[\begin{array}{cc}1& b\\ b& 0\end{array}\right]$
				$0\oplus 1$
8		$0\oplus d$		$\left[\begin{array}{cc}0& b\\ b& 0\end{array}\right]$		$0\oplus d$		$0\oplus 1$
		$\left[\begin{array}{cc}0& b\\ b& 0\end{array}\right]$						$1\oplus 0$
		$a\oplus 0$
7		$0_2$		$0_2$		$0_2$		$\left[\begin{array}{cc}0& b\\ b& 0\end{array}\right]$
6								$0_2$
9	$I_2$	$a\oplus d$, $a<d$	$1\oplus -1$	$a\oplus d$, $a<d$	$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$	$1\oplus d{e}^{i\theta }$	$1\oplus 0$	$a\oplus 1$
						$\left[\begin{array}{cc}0& b\\ b& 1\end{array}\right]$
8		$d_0\oplus d$		$d_0\oplus d$		$1\oplus 0$		$0\oplus 1$
				$\left[\begin{array}{cc}0& b\\ b& 0\end{array}\right]$				$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
6					$0_2$	$I_2$
5		$0_2$		$0_2$				$a\oplus 0$
4						$1\oplus 0$		$0_2$
0						$0_2$

Here $0<\tau <1$, $0<\theta <\pi$, a, b, d>0, $d_0\in \{0,d\}$, $r\ge 0$, $\zeta \in \mathbb{C}$, $0\le \varphi <\pi$.

Figure 1. The closure graph for the action (7(7) $\begin{array}{rl}& {\mathrm{\Psi }}_1:\left((c,P),(A,B)\right)\mapsto c{P}^{\ast }AP,P\in G{L}_2\left(\mathbb{C}\right),c\in S^1,\end{array}$(7) ); $0<\theta <\pi$, $0<\tau <1$. (Orbits at the same horizontal level have the same dimension and these are indicated on the right.)

Table 2. Given $E=c{P}^{\ast }AP-\tilde{A}$, $c\in S^1$, $P=\left[\begin{array}{c}xy\\ uv\end{array}\right]\in G{L}_2\left(\mathbb{C}\right)$, $E\in {\mathbb{C}}^{2\times 2}$, the moduli of the expressions listed in the fourth column are bounded from above by $\nu \left|E\right|$. (The constant $\nu >0$ depends only on A, $\tilde{A}$.)

	$\tilde{A}$	A
C1	$\alpha \oplus 0$	$1\oplus e^{i\theta }$	$|x{|}^2+e^{i\theta }|u{|}^2-c^{-1}\alpha$, $y^2,v^2$	$\alpha \in \{0,1\}$, $0\le \theta <\pi$
C2	$\alpha \oplus 0$	$\left[\begin{array}{cc}0& 1\\ \tau & 0\end{array}\right]$	$\overline{y}v$, $\overline{x}v$, $\overline{u}y$, $(1+\tau )\mathrm{Re}\left(\overline{x}u\right)+i(1-\tau )\mathrm{Im}\left(\overline{x}u\right)-\frac{\alpha }{c}$	$0\le \tau <1,\alpha \in \{0,1\}$
C3	$\left[\begin{array}{cc}\alpha & \text{β}\\ \text{β}& \omega \end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$	$2\mathrm{Re}\left(\overline{y}v\right)-(-1{)}^k\omega$, $2\mathrm{Re}\left(\overline{x}u\right)-(-1{)}^k\alpha$	$k\in \mathbb{Z}$; $\alpha =\omega =0,\text{β}=1\mathrm{o}\mathrm{r}$
			$(\overline{x}v+\overline{u}y)-(-1{)}^k\text{β}$	$\text{β}=0,\alpha \in \{0,1\},\omega \in \{0,\pm \alpha \}$
C4	$\alpha \oplus 0$	$\left[\begin{array}{cc}0& 1\\ 1& i\end{array}\right]$	$\overline{x}v+\overline{u}y$, $\overline{u}v$, $\mathrm{Re}\left(\overline{y}u\right)$, $v^2$, $2\mathrm{Re}\left(\overline{x}u\right)+i|u{|}^2-\frac{\alpha }{c}$	$\alpha \in \{0,1\}$
C5	$1\oplus e^{i\theta }$	$1\oplus e^{i\theta }$	$u^2,y^2$, $|x{|}^2-1$, $|v{|}^2-1$	$0<\theta <\pi$
C6	$\left[\begin{array}{cc}\alpha & \text{β}\\ \text{β}& \omega \end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ 1& i\end{array}\right]$	$2\mathrm{Re}\left(\overline{x}u\right)-(-1{)}^k\alpha$, $2\mathrm{Re}\left(\overline{y}v\right)-(-1{)}^k\mathrm{Re}(\omega )$	$k\in \mathbb{Z}$; $\text{β}=1,\alpha =0,\omega \in \{0,i\}$
			$\overline{x}v+\overline{u}y-(-1{)}^k\text{β}$, $u^2$, $|v{|}^2-(-1{)}^k\mathrm{Im}\left(\omega \right)$	$\mathrm{o}\mathrm{r}\text{β}=0,-\omega =\alpha \in \{0,1\}$
C7	$\left[\begin{array}{cc}0& 1\\ \tau & 0\end{array}\right]$	$\left[\begin{array}{cc}0& 1\\ \tau & 0\end{array}\right]$	$\overline{x}u$, $\overline{y}v$, $\overline{y}u$, $\overline{v}x-c^{-1}$	$0\le \tau <1$
			$c-(-1{)}^k$	$0<\tau <1$, $k\in \mathbb{Z}$
C8	$\alpha \oplus \omega$	$1\oplus \sigma$	$\left(\right|x{|}^2+\sigma \left|u{|}^2\right)-c^{-1}\alpha$, $\overline{x}y+\sigma \overline{u}v$	$\alpha =1,\omega \in \{\sigma ,0\}\mathrm{o}\mathrm{r}\alpha =\omega =0$
			$\left(\right|y{|}^2+\sigma \left|v{|}^2\right)-c^{-1}\omega$	$\sigma \in \{1,-1\}$
			$c-(-1{)}^k$ ($k=0\mathrm{f}\mathrm{o}\mathrm{r}\sigma =1$)	$\sigma =\omega \in \{1,-1\}$, $k\in \mathbb{Z}$
C9	$\alpha \oplus 0$	$1\oplus 0$	$y^2$, $|x{|}^2-\alpha$	$\alpha \in \{0,1\}$
			c−1	$\alpha =1$, $\left|E\right|\le \frac{1}{2}$
C10	$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$	$1\oplus -1$	$\overline{x}y-\overline{u}v-(-1{)}^k$, $|x{|}^2-|u{|}^2$, $|y{|}^2-|v{|}^2$

Figure 2. All nontrivial paths $(\tilde{A},\tilde{B})\to (A,B)$ with nontrivial $(\tilde{A},\tilde{B})\ne (1\oplus 0,\tilde{a}\oplus 0)$ for $\tilde{a}\ge 0$ in the closure graph for the action (6(6) $\mathrm{\Psi }:\left((c,P),(A,B)\right)\mapsto (c{P}^{\ast }AP,P^{T}BP),P\in G{L}_2\left(\mathbb{C}\right),c\in S^1.$(6) ). The dimensions of orbits are indicated on the right.