  We consider a Cartier divisor L on a d-dimensional complex projective variety
X. It is well-known that the dimensions of the cohomomology groups
H^i(X,O_X(mL)) grow at most like m^d, and it is natural to ask when one of
these actually has maximal growth. For i = 0, this happens by definition
exactly when L is big. Here we focus on the question of when one or more of the
higher cohomology groups grows maximally. Our main result is that if one
considers also small perturbations of the divisor in question, then the maximal
growth of higher cohomology characterizes non-ample divisors. This criterion
can also be phrased in terms of the vanishing of continuous functions on the
Neron-Severi space of X that measure the growth of higher cohomology groups.
