  We give some bounds on the anticanonical degrees of Fano varieties with
Picard number 1 and mild singularities, extending results of Koll\'ar et al.
from the early 90's and improving them even in the smooth case. The proof is
based on a study of positivity properties of sheaves of differential operators
on ample line bundles, and avoids the use of rational curves and
bend-and-break. This note is a self-contained exposition of the main ideas of
math.AG/9811022
