TITLE: Intersections of complex submanifolds in $\mathbb{C}^N$
QUESTION [5 upvotes]: This is an exercise from Gromov's Partial differential relations. (page 5) 
Let $V$ and $V'$ be two closed complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ intersect if the following sets are compact for all k. 
$V_k = \{(v,v') \subset V\times V'|dist(v,v') \leq k$}.  
I was looking for a differential geometry approach to solving this problem along the lines of Theorem 2 of Frankel(1961) but anything would do.

REPLY [9 votes]: By compactness, there is a pair of points at the minimal distance. On the other hand, any positive distance can be made smaller (e.g., cf. the proof of Lefschetz theorem in Milnor's "Morse theory": the critical points of the distance function to a complex manifold always have positive index).