my $path_base = 'C:\Users\darkblack\Desktop\A\.';
...
my $dir = $path_base . $directory_name;

##</code><code>##

$ pwd
/Users/ken/tmp
$ ls -al .fred
ls: cannot access '.fred': No such file or directory
$ perl -e 'my $x = q{/Users/ken/tmp/.}; my $y = q{fred}; mkdir $x . $y'
$ ls -al .fred
total 0
drwxr-xr-x  2 ken staff  68 Dec  4 18:27 .
drwxr-xr-x 19 ken staff 646 Dec  4 18:27 ..