Higher cohomology of divisors on a projective variety -- Tommaso de Fernex 

<p>The purpose of this talk is to present an ampleness criterion for line
bundles on projective varieties using growth rate of higher cohomological
dimensions. We consider a Cartier divisor D on a d-dimensional complex
projective variety X. It is well-known that the dimensions of the
cohomomology groups H^i(X,O_X(mD)) 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 D 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 certain continuous functions,
called "asymptotic cohomological functions", on the Neron-Severi space of
X. This is joint work with Alex Kuronya and Robert Lazarsfeld.