This paper shows that a natural “abundance” statement can fail for certain three-dimensional complex spaces in the Kähler category. In algebraic geometry the abundance conjecture says that when a canonical divisor is numerically nonnegative (nef), then some multiple of it should give a map (technical term: be semiample). Here “nef” means the divisor has nonnegative intersection with every curve, and “semiample” means some positive multiple defines a global map. The author builds a compact Kähler threefold that breaks this expected behaviour for semi log canonical (slc) singularities.

Concretely, the paper constructs an irreducible compact Kähler slc threefold X with K_X nef, whose normalization μ:(X̃, D̃)→X satisfies κ(X̃, K_{X̃}+D̃)=0 (the log Kodaira dimension is zero), but K_X is not semiample. In fact the author shows that for every sufficiently large and divisible positive integer m, the space of global sections H^0(X, mK_X) is zero. This counterexample is presented in Section 6 of the paper.

On the positive side, the author proves that abundance does hold for a milder class of singularities called semi-divisorial log terminal (sdlt) pairs. For any compact Kähler sdlt threefold (X, Δ) with K_X+Δ nef, the sheaf O_X(m(K_X+Δ)) is globally generated for all large divisible m, which is another way to say K_X+Δ is semiample. This result appears as Theorem 1.3 and is proved in Section 5. The sdlt condition roughly means the singularities are more controlled and the components of the normalization match the original components.

The paper also gives a hybrid positive result for general slc pairs. If (X, Δ) is a compact Kähler slc threefold with K_X+Δ nef, and if every component of the normalization has positive log Kodaira dimension κ>0, then K_X+Δ is semiample (Theorem 1.5). This shows the counterexample is tied to the case where the normalization has κ=0. The author explains that one obstruction in the Kähler setting is the failure of a classical finiteness property for pluricanonical representations on some non-algebraic K3 surfaces. To work around that, new notions called minimally admissible and minimally preadmissible sections are introduced and used to adapt Fujino’s earlier strategy.