NOTIONS OF NUMERICAL IITAKA DIMENSION DO NOT COINCIDE

Let X be a smooth projective variety. The Iitaka dimension of a divisor D is an important invariant, but it does not only depend on the numerical class of D. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseuodoeffective R-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R-divisor D+ for which h⁰(X, ⌊mD+⌋ + A) is bounded above and below by multiples of m³/² for any sufficiently ample A.