  The lower dimensional Busemann-Petty problem asks whether origin-symmetric
convex bodies in R^n with smaller volume of all k-dimensional sections
necessarily have smaller volume. The answer is negative for k>3. The problem is
still open for k=2,3. We study this problem in the complex hyperbolic n-space
and prove that the answer is affirmative only for sections of complex dimension
one and negative for sections of higher dimensions.
