Abundance conjecture

To describe a map using projective coordinates, its formulas must not all become zero at any point. Can canonical differential-form data that is numerically nonnegative along every curve always supply such everywhere-defined formulas? Abundance asks this for complex projective algebraic shapes with mild singularities and a boundary carrying nonnegative rational weights. Numerical nonnegativity is the starting condition; enough global sections to make a map is the missing conclusion.[1]

Can canonical data that passes every numerical curve test always define a map at every point of a minimal model? A map into projective space uses global sections as coordinates; a common zero would leave the map undefined there. Abundance asks for sections without that obstruction, not merely a nonzero section somewhere.

For a complex projective klt pair $(X,\Delta)$, with $\Delta$ an effective $\mathbb Q$-divisor, does the nef divisor $K_X+\Delta$ become semiample, meaning that $\lvert m(K_X+\Delta)\rvert$ is basepoint-free for some positive integer $m$ making it Cartier? Dimension is unrestricted; the singularity and rational-boundary hypotheses remain part of the target.[1]

The implication is known for curves, surfaces and projective threefolds. Higher-dimensional criteria cover additional special conditions on the canonical data. They do not give the needed global sections for every allowed pair in arbitrary dimension.[1] [2]

Known higher-dimensional metric criteria impose conditions such as nonzero holomorphic Euler characteristic and extra positivity of a pulled-back metric, sometimes with lower-dimensional programme assumptions. The Kähler reduction separately assumes projective abundance in its inductive hypotheses. These are useful conditional routes, not a proof for every projective klt pair with nef adjoint divisor.[1] [2]

The minimal-model programme simplifies algebraic shapes, but numerical tests alone do not describe how their points are organized. A globally defined canonical map supplies that organization: when it is nonconstant, points with the same output form fibres over a simpler base. This connects curve-by-curve numerical information to the geometric families used in classification.[2]

Semiampleness makes the Iitaka fibration globally defined on the minimal model, so the growth of canonical sections is represented by an actual geometric map. Where that map is nonconstant, one can organize the variety through its base and fibres, rather than retain only intersection numbers. This is why abundance is part of birational classification; conditional Kähler reductions use that structure but cannot manufacture the missing projective theorem.[1] [2]