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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, if (with respect to complex coordinates $z_{1}, ..., z_{n}$) the function $\frac{\partial^{2}d_{X}}{\partial z_{i} \partial \overline{z}_{j}}$ is holomorphic? (assume that we have alredy choosen a hermitian metric to compute $d_{X}$).

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Which real-valued functions are holomorphic? – Pierre-Yves Gaillard Aug 13 '11 at 10:18
I'm a total novice in $\mathbf{C}^n$, so it may be easy, but: Can we say something about those subsets $X$ for which the squared distance $d_X(z)$ can be written in the form $$d_X(z)=f(z)\overline{f(z)}$$ for some holomorphic function $f$. Works when $X$ is a point and $n=1$. Works in higher $n$, if we also allow vector valued functions $f$ (and want to write $d_X$ as the squared hermitian norm. Any other way to make this question more interesting? – Jyrki Lahtonen Aug 13 '11 at 13:34
up vote 4 down vote accepted

No. Take $n=1$, $M=\mathbb C$ and $X=\{|z|=1\}$. Then $$ d(z)=\left|z-\frac z{|z|}\right|^2=\left(z-\frac z{\sqrt{z\bar z}}\right) \left(\bar z-\frac {\bar z}{\sqrt{z\bar z}}\right)= z\bar z -2\sqrt{z\bar z}+1 $$ and $$ \frac{\partial^{2}d(z)}{\partial z \partial \bar{z}}= 1-\frac {1}{2\sqrt{z\bar z}}. $$

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It makes the question look kind of sad if the first metric that comes in mind doesn't satisfy OP's wishes. I believe he should've worked a little bit more... – Patrick Da Silva Aug 13 '11 at 12:05

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