Chart in $1$-parameter family of Lagrangians in a Kähler manifold

Question

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}$ be a analytically varying $1$-parameter family of real-analytic Lagrangian submanifolds of $X$ such that $L_{0} = L$ (here one should think of the parameter $t$ as sufficiently small) and not intersection each other, i.e. $L_{t} \cap L_{t'} = \emptyset$, $\forall t,t'$. Furthermore, let $p\in L$ be a point.

Question: Does there exists a holomorphic chart $\varphi : U \rightarrow V$ around $p$, where $U \subset X$, $V \subset \mathbb{C}^{n}$ and $p \in U$ such that:

1. $V \cap (\mathbb{R}^{n} \times \{0\}) \not= \emptyset$ (here we view $\mathbb{C}^{n} = \mathbb{R}^{n} \oplus i\mathbb{R}^{n}$);

2. $\forall t$ we have $\varphi(U\cap L_{t}) = V\cap (\mathbb{R}^{n}\times \{t\} \times \underbrace{\{0\}}_{\in \mathbb{R}^{2n-1}})$?

If this is not true (or you think that it might not be true) do you have any idea if there exists something similar (in any sense) for a non-intersection $1$-parameter family of Lagrangian submanifolds?

Answer 1

In any holomorphic chart, real analytic submanifolds remain real analytic. If your Lagrangian manifolds are not real analytic, they cannot become real analytic in holomorphic coordinates. In fact, you cannot even arrange that they lie in a real analytic hypersurface, let alone become these particular real analytic submanifolds.

Besides real analyticity, there is an invariantly defined section of the normal bundle of each Lagrangian manifold $L_t$, which, at each point of $L_t$, gives the normal component of the velocity of that point as $t$ varies. The tangential component is not defined, as $L_t$ is only defined as a submanifold, unparameterized. If this normal velocity field vanishes at some point of some $L_t$, then no such holomorphic coordinates can exist.

I think that if that normal velocity field is nonzero, and the family $L_t$ is real analytic, then you can get such holomorphic coordinates, but I have no tried to work the details out yet.