Math Conjectures

Abundance Conjecture

OPENLandmarkConjectureProposed c. 1985 · Standard version

Canonical statement

Let (X,Δ)(X,\Delta) be a projective Kawamata-log-terminal pair over C\mathbb C, with KX+ΔK_X+\Delta a Q\mathbb Q-Cartier Q\mathbb Q-divisor. If KX+ΔK_X+\Delta is nef, then it is semiample: some positive integral multiple is Cartier and base-point-free.
View source LaTeX
Let \((X,\Delta)\) be a projective Kawamata-log-terminal pair over \(\mathbb C\), with \(K_X+\Delta\) a \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisor. If \(K_X+\Delta\) is nef, then it is semiample: some positive integral multiple is Cartier and base-point-free.
AG-MMP-010:[Kawamata1985Pluricanonical][KollarMori1998Birational][Fujino2011Fundamental]
Algebraic geometryGeometry and topology

Notes

The abundance conjecture concerns the expected final output of the minimal model program over C\mathbb C: for a projective Kawamata log terminal pair (X,Δ)(X,\Delta) with KX+ΔK_X+\Delta nef, it predicts that KX+ΔK_X+\Delta is semiample, so some positive multiple is Cartier and base-point-free. The statement crystallized during the mid-1980s development of the program rather than on a single date, with Kawamata's study of pluricanonical systems on minimal algebraic varieties an early landmark [Kawamata1985Pluricanonical]; its role in the general theory is explained in [KollarMori1998Birational].

Abundance is known in dimension at most three, for varieties of general type, and in many special higher-dimensional situations; the fundamental theorems of the log minimal model program surrounding it have been established in considerable generality [Fujino2011Fundamental].

Settling the remaining cases would complete a central pillar of the birational classification of algebraic varieties, but the hardest ones, namely KX+ΔK_X+\Delta nef and numerically trivial and the intermediate Kodaira-dimension range, remain open in higher dimension.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.