  Using density functional theory calculations including an on-site Coulomb
term, we explore electronic and possibly topologically nontrivial phases in
$3d$ transition metal oxide honeycomb layers confined in the corundum structure
($\alpha$-Al$_2$O$_3$) along the [0001] direction. In most cases the ground
state is a trivial antiferromagnetic Mott insulator, often with distinct
orbital or spin states compared to the bulk phases. With imposed symmetry of
the two sublattices the ferromagnetic phases of Ti, Mn, Co and Ni exhibit a
characteristic set of four bands, two relatively flat and two with a Dirac
crossing at K, associated with the single electron occupation of $e_{g}'$ (Ti)
or $e_{g}$ (Mn, Co, Ni) orbitals. Our results indicate that the Dirac point can
be tuned to the Fermi level using strain. Applying spin-orbit coupling (SOC)
leads to a substantial anomalous Hall conductivity with values up to 0.94
$e^2/h$. Moreover, at $a_{Al_2O_3}$=4.81\AA\ we identify a particularly strong
effect of SOC with out-of-plane easy axis for
($Ti_2$O$_3$)$_1$/(Al$_2$O$_3$)$_5$(0001) which stabilizes dynamically the
system. Due to the unusually high orbital moment of -0.88$\mu_{\rm B}$ that
nearly compensates the spin moment of 1.01$\mu_{\rm B}$, this system emerges as
a candidate for the realization of the topological Haldane model of spinless
fermions. Parallels to the perovskite analogs
(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$(111) are discussed.
