Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, **$Y$ is not smooth**. At a point $y \in Y$, a generic, sufficiently small polydisc $\mathbb{D}^k \ni y$ will satisfy $\mathbb{D}^k \cap Y = y$. I would like to do this continuously along the $\mathbb{C}^d$.

For $p \in \mathbb{C}^d$, does there exist an analytic neighborhood $p \in U \subset \mathbb{C}^d$, and a subvariety $\widetilde{U} \cong U \times \mathbb{D}^k \subset \mathbb{C}^N$ such that $\widetilde{U} \cap Y = U$?

(I am not really sure what the right tag for this question is. Actually if someone would clue me in as to where to find some basic treatment of whatever subject this question belongs to, that would be great.)