  We consider supersymmetric AdS$_3 \times Y_7$ and AdS$_2 \times Y_9$
solutions of type IIB and $D=11$ supergravity, respectively, that are
holographically dual to SCFTs with $(0,2)$ supersymmetry in two dimensions and
$\mathcal{N}=2$ supersymmetry in one dimension. The geometry of $Y_{2n+1}$,
which can be defined for $n\ge 3$, shares many similarities with
Sasaki-Einstein geometry, including the existence of a canonical R-symmetry
Killing vector, but there are also some crucial differences. We show that the
R-symmetry Killing vector may be determined by extremizing a function that
depends only on certain global, topological data. In particular, assuming it
exists, for $n=3$ one can compute the central charge of an AdS$_3 \times Y_7$
solution without knowing its explicit form. We interpret this as a geometric
dual of $c$-extremization in $(0,2)$ SCFTs. For the case of AdS$_2 \times Y_9$
solutions we show that the extremal problem can be used to obtain properties of
the dual quantum mechanics, including obtaining the entropy of a class of
supersymmetric black holes in AdS$_4$. We also study many specific examples of
the type AdS$_3\times T^2 \times Y_5$, including a new family of explicit
supergravity solutions. In addition we discuss the possibility that the $(0,2)$
SCFTs dual to these solutions can arise from the compactification on $T^2$ of
certain $d=4$ quiver gauge theories associated with five-dimensional
Sasaki-Einstein metrics and, surprisingly, come to a negative conclusion.
