  We relate across dimensions BPS attractors of black strings and black holes
of various topology in gauged supergravities with nontrivial scalar potential.
The attractors are of the form AdS$_{2, 3} \times \Sigma^{2, 3}$ in 4, 5, and 6
dimensions, and can be generalized to some higher dimensional analogs. Even
though the attractor geometries admit standard Kaluza-Klein and Scherk-Schwarz
reductions, their asymptotic AdS spaces in general do not. The resulting lower
dimensional objects are black holes with runaway asymptotics in supergravity
theories with no maximally symmetric vacua. Such classes of solutions are
already known to exist in literature, and results here suggest an
interpretation in terms of their higher-dimensional origin that often has a
full string theory embedding. In a particular relevant example, the relation
between 5d Benini-Bobev black strings arXiv:1302.4451 and a class of 4d
Cacciatori-Klemm black holes arXiv:0911.4926 is worked out in full detail,
providing a type IIB and dual field theory description of the latter solutions.
As a consistency check, the Cardy formula for the field theory is shown to
match the Bekenstein-Hawking entropy for horizon topology of any genus.
