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=Abundance Conjecture=

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In algebraic geometry, the abundance conjecture is a conjecture of birational geometry. It predicts that if the canonical bundle $$K_X$$ of a projective variety $$X$$ is positive in an appropriate sense, then $$K_X$$ has an "abundance" of sections.

Statement of the conjecture
The simplest form of the conjecture is as follows.

Abundance conjecture. Let $$X$$ be a smooth projective variety. If $$K_X$$ is nef, then it is semi-ample: that is, the line bundle $$mK_X$$ is globally generated, or equivalently the linear system $$|mK_X|$$ is basepoint-free, for some $$m>0$$.

In other words, if $$K_X$$ is nef, then its section are "abundant" enough to determine a morphism of $$X$$ to projective space.

Log abundance
The conjecture also has a "logarithmic" version, as is common in birational geometry.

Log abundance conjecture. Let $$X$$ be a projective variety. Suppose $$\Delta$$ is an effective divisor on $$X$$ such that the pair $$(X,\Delta)$$ is log canonical. If $$K_X+\Delta$$ is nef, then it is semi-ample.

Here log canonical is one of the classes of singularites of pairs encountered in minimal model theory. Roughly speaking, it is conjecturally the largest class of singularities which is closed under the operations of the log minimal model program.

History and status
Surfaces.

For three-dimensional varieties in characteristic 0, the conjecture was proved by Miyaoka and Kawamata. The logarithmic version was proved by Keel, Matsuki, and McKernan.

Statement of the conjecture
Let f: X→Z be an algebraic fibre space with X and Z smooth projective varieties, and let F be a general fibre of f. Then &kappa;(X) ≥ &kappa;(Z)+&kappa;(F), where &kappa; denotes the Kodaira dimension.

History and status

 * Iitaka


 * Kawamata

=Gorenstein and Cohen–Macaulay schemes=

Explain cohomological significance: Gorenstein => dualising line bundle, Cohen–Macaulay => dualising sheaf