i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ sufficiently near $X$ this function is smooth. My quaestion is: Is (with respect to complex coordinates $z_{1}, ..., z_{n}$) the function $\partial^{2} d_{X}/\partial d_{X} / \partial z_{i} \partial \overline{z}{j}$ bar z_{j}$ is holomorphic? (assume that we have alredy choosen a hermitian metric to compute $d{X}$).d_{X}$).

i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ sufficiently near $X$ this function is smooth. My quaestion is: Is (with respect to complex coordinates $z_{1}, ..., z_{n}$) the function $\partial^{2} d_{X}/\partial z_{i} \partial \overline{z}{j}$ is holomorphic? (assume that we have alredy choosen a hermitian metric to compute $d{X}$).