i have the following problem. Let $M$ be a complex ndim manifold and $X \subset M$ be a ndim 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 $\partial^{2} d_{X} / \partial z_{i} \partial \bar z_{j}$ is holomorphic? (assume that we have alredy choosen a hermitian metric to compute $d_{X}$).
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Take the most simple example $M=\mathbb C$ and $X$ the unit circle. Then $d_X(z)=z\frac{z}{z}^2$ is not holomorphic (so as its Laplacian $i\partial \bar \partial( d_X)$), as you can easily see by expanding the expression. 

