# Intersections of complex submanifolds in $\mathbb{C}^N$

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.

• I am confused. Take the lines $x=0$ and $x=1$ in $\mathbb{C}^2$. There is no compactness of course, but still the distance function has minimum value $1$. Why does your argument not apply? – abx Jan 14 '14 at 11:33
• @AlexDegtyarev You mentioned On the other hand, any positive distance can be made smaller and you gave a reference to Milnor's Morse. But after skimming through the proof of the Lefschetz theorem, there is no mention of the distance function having a positive index. There is a bound though, and that is what is used in that book. Also in the answer there is no mention of the fact that $V$ and $V'$ have complementary dimension. – bellztoll Jan 16 '14 at 8:30