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Using density functional theory calculations with a Hubbard $U$, we exploretopologically nontrivial phases in $X_2$O$_3$ honeycomb layers with $X=$ $4d$and $5d$ cation inserted in the band insulator $\alpha$-Al$_2$O$_3$ along the[0001]-direction. Several promising candidates for quantum anomalous Hallinsulators (QAHI) are identified. In particular, for $X$=Tc and Pt spin-orbitcoupling (SOC) opens a gap of 54 and 59 meV, respectively, leading to Cherninsulators (CI) with $C$=--2 and --1. The nature of different Chern numbers isrelated to the corresponding spin textures. The Chern insulating phase issensitive to the Coulomb repulsion strength: $X$=Tc undergoes a transition froma CI to a trivial metallic state beyond a critical strength of $U_c$ =2.5 eV. Acomparison between the isoelectronic metastable FM phases of $X$=Pd and Ptemphasizes the intricate balance between electronic correlations and SOC: whilethe former is a trivial insulator, the latter is a Chern insulator. Inaddition, $X$=Os turns out to be a FM Mott insulator with an unpaired electronin the $t_{2g}$ manifold where SOC induces an unusually high orbital moment of0.34 $\mu_{\rm B}$ along the $z$-axis. Parallels to the $3d$ honeycomb corundumcases are discussed.

Interaction-driven spin-orbit effects and Chern insulating phases in corundum-based $4d$ and $5d$ oxide honeycomb lattices