Vũ Ngọc’s conjecture on focus-focus singular fibers with multiple pinched points

Abstract

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing $k \geqslant 1$ focus-focus critical points. Our proof recovers the classification for $k=1$ which was known prior to this paper. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a $(\mathbb {Z}_2 \times D_k)$-action, where $D_k$ is the kth dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold $(M,\omega )$ near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.

Appendix A. Technical results on flat functions

Recall that ${{\,\textrm{Taylor}\,}}_0[f] \in \mathbb {R}[[\textrm{T}^*_0U]]$ denotes the Taylor series of $f :U \rightarrow \mathbb {R}$ at the origin for $U \in \mathcal {N}(\mathbb {R}^2, 0)$. Note how the Taylor series, as a formal power series, depends on the choice of a basis of $\textrm{T}^*_0U$. Note that the Taylor series only depends on the germ of the function.

Definition A.1

We call f a flat function at 0, or the function f is flat at 0, if ${\text {Taylor}}_0[f] = 0$. Denote by $\mathcal {O}(c^{\infty })$ the space of flat functions at 0.

Note that, by the Faà di Bruno’s formula, the Taylor series of the composition of smooth maps is the composition of their Taylor series. This is why the definition of flat functions is independent of the choice of the basis of $\textrm{T}^*_0U$.

We will use the multi-index notations in Lemmas A.2 and A.3. A multi-index j is a pair $(j_1, j_2)$ where $j_1, j_2 \in \mathbb {Z}_{\geqslant 0}$. We use $\left|j\right| = j_1 + j_2$. If $c = (c_1, c_2) \in \mathbb {R}^2$ then $c^j = c_1^{j_1} c_2^{j_2}$. If f is a function on an open subset of $\mathbb {R}^2$ then $\partial ^j f = \frac{\partial ^{\left|j\right|} f}{\partial c_1^{j_1} \partial c_2^{j_2}}$.

Lemma A.2

Let $m \in \mathbb {Z}_{\geqslant 0}$. Let $g_j :\mathbb {R}^2 \rightarrow \mathbb {R}$ be a smooth function for multi-index j with $\left|j\right| = m$ and let $g(c) = \sum _{\left|j\right| = m} g_j(c) c^j$. Then the function $g \ln \left|\cdot \right|:\mathbb {R}^2_\textrm{r}\rightarrow \mathbb {R}$ can be extended to a $\textrm{C}^{m-1}$-function on $\mathbb {R}^2$ if $m \geqslant 1$. Furthermore, the extension is $\textrm{C}^m$ if and only if $g_j(0) = 0$ for all j.

Proof

Let $L_m$, $m \in \mathbb {Z}_{\geqslant 0}$, be the $\textrm{C}^\infty (\mathbb {R}^2)$-vector space spanned by functions of the form $\left( \mathbb {R}^2_\textrm{r}\ni c \mapsto c^j \ln \left|c\right|\right)$ for which j is a multi-index with $\left|j\right| \geqslant m$, extended onto $\mathbb {R}^2$ by zero. Let $Q_m$, $m \in \mathbb {Z}$, be the $\textrm{C}^\infty (\mathbb {R}^2)$-vector space spanned by functions of the form $\left( \mathbb {R}^2_\textrm{r}\ni c \mapsto \frac{c^j}{\left|c\right|^{m_0}}\right)$ for which j is a multi-index, $m_0 \in \mathbb {Z}_{\geqslant 0}$, and $\left|j\right| \geqslant m_0 + m$, extended onto $\mathbb {R}^2$ by zero. By direct calculations:

$$\begin{aligned} \frac{\partial }{\partial c_1} \left( c^j \ln \left|c\right|\right)&= j_1 c_1^{j_1-1} c_2^{j_2} \ln \left|c\right| + \frac{c_1^{j_1+1} c_2^{j_2}}{\left|c\right|^2}, \\ \frac{\partial }{\partial c_1} \left( \frac{c^j}{\left|c\right|^{m_0}}\right)&= j_1 \frac{c_1^{j_1-1} c_2^{j_2}}{\left|c\right|^{m_0}} - m_0 \frac{c_1^{j_1+1} c_2^{j_2}}{\left|c\right|^{m_0+2}}, \end{aligned}$$

which implies that if $f \in L_m$ then $\frac{\partial }{\partial c_1} f, \frac{\partial }{\partial c_2} f \in L_{m-1} + Q_{m-1}$ for $m \geqslant 1$, and if $f \in Q_m$ then $\frac{\partial }{\partial c_1} f, \frac{\partial }{\partial c_2} f \in Q_{m-1}$ for $m \in \mathbb {Z}$. Therefore, $Q_0 \subseteq \textrm{B}_0$ the set of functions on $\mathbb {R}^2$ bounded in a neighborhood of the origin, $L_1, Q_1 \subseteq \textrm{C}^0$ and then $L_m, Q_m \subseteq \textrm{C}^{m-1}$ for $m \geqslant 1$.

We aim to find $L_0 \cap \textrm{B}_0$ and let $f = h \ln \left|\cdot \right|\in L_0$ for some $h \in \textrm{C}^\infty$. On one hand, if $f \in L_0 \cap \textrm{B}_0$, then the boundedness requires that $h(0) = 0$. On the other hand, if $h(0) = 0$, then $h(c) = h_1 c_1 + h_2 c_2$ for some $h_1, h_2 \in \textrm{C}^\infty$ and then $f \in L_1 \subseteq \textrm{C}^0$. Therefore, $L_0 \cap \textrm{B}_0 = L_1$.

In particular, given $g(c) = \sum _{\left|j\right| = m} g_j(c) c^j$, we have $g \ln \left|\cdot \right|\in L_m \subseteq \textrm{C}^{m-1}$ if $m \geqslant 1$, and for any multi-index j with $\left|j\right| = m$, we have

$$\begin{aligned} \partial ^j \left( g(c) \ln \left|c\right|\right) \in j!\, g_j(c) \ln \left|c\right| + L_1 + Q_0. \end{aligned}$$

On one hand, $g \ln \left|\cdot \right|\in \textrm{C}^m$ requires that $\partial ^j \left( g(c) \ln \left|c\right|\right) \in \textrm{B}_0$, and hence $g_j(0) = 0$. On the other hand, $g_j(0) = 0$ for every multi-index j with $\left|j\right| = m$ implies that $g \ln \left|\cdot \right|\in L_{m+1} \subseteq \textrm{C}^m$. $\square$

Lemma A.3

For a smooth function f on $\mathbb {R}^2$, the function $f\ln \left|\cdot \right|$ on $\mathbb {R}^2_\textrm{r}$ can be smoothly extended onto $\mathbb {R}^2$ only when f is flat. If f is flat, the extension of $f\ln \left|\cdot \right|$ is also flat.

Proof

Using Taylor expansion of f, for any $m \in \mathbb {N}$, there exist smooth functions $g_j :\mathbb {R}^2 \rightarrow \mathbb {R}$ for all multi-indices j with $\left|j\right| = m+1$ such that

$$\begin{aligned} f(c) = \sum _{\left|j\right| = 0}^m \frac{1}{j!} \partial ^j f(0) c^j + \sum _{\left|j\right| = m+1} \frac{1}{j!} g_j(c) c^j. \end{aligned}$$

(A.1)

By Lemma A.2 and (A.1), $f\ln \left|\cdot \right|\in \textrm{C}^m$ if and only if $\partial ^j f(0) = 0$ for any multi-index j with $\left|j\right| \leqslant m$. Therefore, $f\ln \left|\cdot \right|\in \textrm{C}^\infty$ if and only if $f \in \mathcal {O}(c^{\infty })$.

Note that $\ln \left|c\right| \in \mathcal {O}(\left|c\right|^{-1})$. If $f \in \mathcal {O}(c^{\infty })$, then for any $m \in \mathbb {N}$, there exist smooth functions $g_j :\mathbb {R}^2 \rightarrow \mathbb {R}$ for all multi-index j with $\left|j\right| = m+1$, such that

$$\begin{aligned} f(c) \ln \left|c\right| = \sum _{\left|j\right| = m+1} \frac{1}{j!} g_j(c) c^j \ln \left|c\right| \in \mathcal {O}(\left|c\right|^{m}). \end{aligned}$$

Hence $f\ln \left|\cdot \right|\in \mathcal {O}(c^{\infty })$. $\square$

Lemma A.4

If g is a flat function on $\mathbb {R}^2$ and h is a smooth function on $\mathbb {R}^2_\textrm{r}$ that satisfies

$$\begin{aligned} \forall \text {multi-index}~j~\exists m_j \in \mathbb {Z}\text { such that } \lim _{c \rightarrow 0} \left|c\right|^{m_j} \left|\partial ^j h(c)\right| = 0, \end{aligned}$$

(A.2)

then $f = gh$ on $\mathbb {R}^2_\textrm{r}$ has a smooth extension $\tilde{f}$ on $\mathbb {R}^2$.

Proof

We calculate the partial derivatives of f for a multi-index j and $m \in \mathbb {Z}$:

$$\begin{aligned} \left|c\right|^m \left|\partial ^j f(c)\right| \leqslant \sum _{0 \leqslant \ell \leqslant j} \left( {\begin{array}{c}j\\ \ell \end{array}}\right) \left|c\right|^{m-m_\ell } \left|\partial ^{j-\ell } g(c)\right| \cdot \left|c\right|^{m_\ell } \left|\partial ^\ell h(c)\right| \rightarrow 0 \end{aligned}$$

(A.3)

as $c \rightarrow 0$. Here, we use the fact that $\partial ^{j-\ell } g$ is a flat function so it is dominated by any power of $\left|c\right|$. Now let $\tilde{f}$ be the extension of f by $\tilde{f}(0) = 0$. Then by (A.3), $\frac{\partial \tilde{f}}{\partial c_1}(0) = \lim _{\delta \rightarrow 0} \frac{f(\delta , 0)}{\delta } = 0$ and $\frac{\partial \tilde{f}}{\partial c_2}(0) = \lim _{\delta \rightarrow 0} \frac{f(0, \delta )}{\delta } = 0$, and then $\tilde{f} \in \textrm{C}^1$. Inductively, we can show that $\tilde{f} \in \textrm{C}^\infty$ and any higher order derivative of $\tilde{f}$ vanishes at the origin. $\square$