  By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for
the volume of all klt varieties with ample canonical class. We show that these
bounds must go to zero extremely fast as the dimension increases, by
constructing a klt $n$-fold with ample canonical class whose volume is less
than $1/2^{2^n}$. These examples should be close to optimal.
  We also construct a klt Fano variety of each dimension $n$ such that
$H^0(X,-mK_X)=0$ for all $1\leq m < b$ with $b$ roughly $2^{2^n}$. Here again
there is some bound in each dimension, by Birkar's theorem on boundedness of
complements, and we are showing that the bound must increase extremely fast
with the dimension.
