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.



A geometric characterization of smooth points of a complex algebraic variety
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 .


