On semipositivity of sheaves of differential operators and the degree of a unipolar Q-Fano variety

We consider normal projective n-dimensional varieties X whose anticanonical divisor class -K is ample and where every Weil divisor is a rational multiple of K. The index i is the largest integer such that K/i exists as a Weil divisor. We show (i) if X has log-terminal singularities, and in addition 1-forms on the smooth part of X are holomorphic on a resolution, then (-K)^n =< (max(in,n+1))^n; (ii) if the tangent sheaf of X is semistable, then (-K)^n =<(2n)^n. The proof is based on some elementary but possibly surprising slope estimates on sheaves of differential operators on plurianticanonical sheaves. Unlike previous arguments in the smooth case (Nadel, Campana, Kollar-Miyaoka-Mori), rational curves and rational connectedness are not used.