Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed Lagrangian submanifold. Furthermore, let $L_{t}$ be a smoothly varying $1$-parameter family of 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?

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?

Let $(X,\omega,J)$ be a complex $n$-dimensional Kaehler manifold ($\omega$ Kaehler form, $J$ complex structure) and $L \subset X$ be a closed Lagrangian submanifold. Furthermore, let $L_{t}$ be a smoothly varying $1$-parameter family of 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?

Cheers, Michael

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed Lagrangian submanifold. Furthermore, let $L_{t}$ be a smoothly varying $1$-parameter family of 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?