Singularity criteria for K-stability of adjoint foliated structures

We prove singularity criteria for the $t$-K-stability of adjoint foliated structures. We first show that K-semistability of adjoint foliated structures implies log canonicity by extending Odaka’s flag ideal characterisation of the mixed Donaldson–Futaki invariant to the adjoint foliated setting. We then prove that adjoint Calabi–Yau foliated structures are K-semistable, and klt ones are K-stable, while log canonical adjoint general type foliated structures are K-stable with respect to the canonical polarisation. We also show that K-semistable adjoint Fano foliated structures are klt. In particular, their ambient varieties are potentially klt and of Fano type.