Notions of numerical Iitaka dimension do not coincide

Author:

Lesieutre John

Abstract

Let X X be a smooth projective variety. The Iitaka dimension of a divisor D D is an important invariant, but it does not only depend on the numerical class of D 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 \mathbb {R} -divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R \mathbb {R} -divisor D + D_+ for which h 0 ( X , m D + + A ) h^0(X,\left \lfloor {m D_+}\right \rfloor +A) is bounded above and below by multiples of m 3 / 2 m^{3/2} for any sufficiently ample A A .

Publisher

American Mathematical Society (AMS)

Subject

Geometry and Topology,Algebra and Number Theory

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