You can get a global handle through a closure. That's more or less ok in OO since it's a method which accesses the handle ;-)

package My;
use DBI;
{
    my $dbh = DBI::connect( . . .); # add error handling
    sub dbh () { $dbh }
}
# call up the handle with My::dbh().
# You don't need to import anything from DBI unless you
# want to see or modify its globals

package My::Collection;
# . . .
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I used an empty prototype to give perl a hint when My::dbh is called without parens.

Note that sharing a dbi connection between objects will break for threaded or forked code. In those cases a private handle should be preferred.