This result should be compared with [Y], Theorem 8. Yau’s result is stronger in many respects (there is no projectivity/rationality assumption and it gives a more precise regularity theory); on the other hand the conditions on the poles of the L.H.S. is less optimal than here.

We expect the projectivity/rationality assumptions to be superfluous. We also expect that a finer regularity theory might be developed for singular KE metrics depending on a finer analysis of the klt singularities involved.