Notions of numerical Iitaka dimension do not coincide

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 .

On the numerical dimension of Calabi-Yau 3-folds of Picard number 2

“…In this section we adapt the estimates for the number of global sections for divisors close to the boundary of Mov(X) from [Les21] to our more general setting.…”

“…Thus, the right-hand-side is bounded from above by a positive constant C 2,1 , whereas as a lower bound we may choose C 1,1 = 1 7 . Next we recall some notation from [Les21] which is used in the following. Let X be as in Assumption 2.1.…”

“…To overcome this problem, there are several definitions of a numerical invariant, the numerical dimension of D, which were commonly believed to be equivalent (see [Nak04,Leh13,Eck16,Fuj20,CP21]). However, in [Les21], Lesieutre shows that the different notions of numerical dimension for a pseudoeffective R-divisor do not coincide. In particular, he shows the existence of a pseudoeffective R-divisor whose numerical dimension κ σ is not an integer.…”

“…Remark 1.2. We refer to [Les21] and the references therein for properties of κ σ and different definitions of the numerical dimension.…”

“…This relies on [LP13] which shows that the Cone conjecture is true for Calabi-Yau varieties of Picard number 2. Then we compute the numerical dimension of a boundary divisor of the movable cone of X following [Les21]. The idea is to pullback a divisor close to the boundary to the fundamental domain (via a birational automorphism).…”