Special Darboux chart for tranverse Lagrangians

Question

In the notes of Fukaya-Oh-Ohta-Ono (Lagrangian intersection Floer theory), Chapter 10 §54.1, it is stated:

Let $L_{1}$ and $L_{2}$ be a pair of oriented Lagrangian submanifolds in $(M, \omega)$ that intersect transversely at $p_{12}$. We fix an ordering of the pair as $(L_{1} , L_{2})$. We can always choose a Darboux chart in a neighborhood $U$ of $p_{12}$, $I \colon U \to V \subset \mathbb{C}^{n}$ so that $I (p_{12}) = 0$, $$ I(L_{1} \cap U) = \mathbb{R}^{n} \cap V, \quad I(L_{2} \cap U) = \sqrt{−1} \mathbb{R}^{n} \cap V.$$ The proof follows from a version of Darboux theorem (see [Theorem 7.1, Wei71]) but strongly relies on the following well-known fact in symplectic linear algebra whose proof we omit.

Lemma 54.1. The linear symplectic group $\text{Sp} (2n)$ acts transitively on the set of transversal pairs of Lagrangian subspaces.

Theorem 7.1 from Weinstein 1971, Symplectic manifolds and their Lagrangian submanifolds, is a result about Lagrangian foliations.

Does someone have a reference for the proof of the first statement and Lemma 54.1? If not, using Theorem 7.1 from Weinstein 1971 requires a Lagrangian foliation, how it such a foliation constructed from just two transverse Lagrangians?

Answer 1

Here is an elementary argument. By Weinstein's tubular neighbourhood theorem we can arrange that locally $L_1$ is the zero section of $T^* \mathbb{R}^n$, and $L_2$ is another Lagrangian intersecting $L_1$ transversely in the fiber over $0$. Trivializing $T^* \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$ with coordinates $(q, p)$, this means that we can write $L_2$ as the graph $\{(\xi(p), p)\}$ of the function $\xi(p) = (\xi_1(p), \ldots, \xi_n(p))$ of the fiber coordinate $p$. The condition that this graph is Lagrangian is equivalent to the requirement that the matrix of partials of $\xi$ is symmetric (this is just a version of the well-known fact that the graph of a 1-form $\eta : X \to T^*X$ is Lagrangian iff $d \eta = 0$).

By the Poincaré lemma we can therefore find a function $H(p) : \mathbb{R}^n \to \mathbb{R}$ so that $\frac{\partial H}{\partial p_i} = -\xi_i(p)$. Viewing $H$ as a function on $T^* \mathbb{R}^n$ the corresponding Hamiltonian vector field is $$ X_H = \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} = - \xi_i(p) \frac{\partial}{\partial q_i}. $$ The time-1 flow under the Hamiltonian isotopy generated by $X_H$ thus defines a symplectomorphism which fixes $L_1$ and takes $L_2$ to the fiber of $T^* \mathbb{R}^n$ over $0$. Identifying my $T^* \mathbb{R}^n$ with your $\mathbb{C}^n$ gives the desired chart.