  We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold
cone singularity that are then further compactified on a Riemann surface
$\Sigma_g$, with an arbitrary partial topological twist for the global $U(1)$
symmetries. This constitutes a rich, infinite class of two-dimensional $(0,2)$
theories. Under the assumption that such a theory flows to a SCFT, we show that
the supergravity formulas for the central charge and $R$-charges of BPS
baryonic operators of the dual AdS$_3$ solution may be computed using only the
toric data of the Calabi-Yau 3-fold and the topological twist parameters. We
exemplify the procedure for both the $Y^{p,q}$ and $X^{p,q}$ 3-fold
singularities, along with their associated dual quiver gauge theories, showing
that the new supergravity results perfectly match the field theory results
obtained using $c$-extremization, for arbitrary twist over $\Sigma_g$. We
furthermore conjecture that the trial central charge $\mathscr{Z}$, which we
define in gravity, matches the field theory trial $c$-function off-shell, and
show this holds in non-trivial examples. Finally, we check our general
geometric formulae against a number of explicitly known supergravity solutions.
