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Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric. Fix an arbitrary point $p\in X$. Let $V(p,\varepsilon)$ denote volume of the intersection of $X$ with the $\varepsilon$-ball centered at $p$, namely $2m$-dimensional Hausdorff measure of the intersection.

Question 1. Whether there exists the limit $m(p):=\lim_{\varepsilon\to +0} \frac{V(p,\varepsilon)}{\omega_{2m} \varepsilon ^{2m}}$, where $\omega_{2m}$ is the volume of the standard $2m$-dimensional Euclidean ball?

Question 2. If the limit in the Question 1 always exists, is it true that this limit $m(p)$ is a natural number?

Question 3. If Question 2 has positive answer, is it true that the point $p$ is smooth if and only if $m(p)=1$?

One may ask the same questions for any Kahler manifold instead of $\mathbb{C}^n$, for example for complex projective space with the Fubini-Study metric. My feeling is that it should not be very important, but I do not have a proof. A reference would be very helpful.

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    $\begingroup$ Note that your $X$ is minimal. Therefore, by monotonicity formula, your limit exists and it has to be $\ge 1$. $\endgroup$
    – Anton Petrunin
    Commented Jun 29, 2014 at 13:02

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The answer to all three of your questions is yes.See the book by E M Chirka titled Complex Analytic Sets pages 189,190 and 120 .These questions are local so this is true on Kahler manifolds .

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