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

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?

• What do you mean by "zero velocity"?
– MicB
Mar 22, 2020 at 9:49
• How should one apply here implicite function therem to obtain such a holomorphic chart?
– MicB
Mar 22, 2020 at 9:49
• Can you provide more details, please?
– MicB
Mar 22, 2020 at 10:55

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.