Higher cohomology of divisors on a projective variety

Abstract

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.