Assuming the non-vanishing conjecture, the abundance conjecture holds for all smooth projective varieties of numerical dimension at most one; unconditionally, it holds in dimension at most five whenever κ≥0 and ν≤1.
Birkar, On existence of log minimal models, Compos
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Non-vanishing implies numerical dimension one abundance
Assuming the non-vanishing conjecture, the abundance conjecture holds for all smooth projective varieties of numerical dimension at most one; unconditionally, it holds in dimension at most five whenever κ≥0 and ν≤1.