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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.

References

• Kawamata, Yujiro (1992). "Abundance theorem for minimal threefolds". Invent. Math. 108 (2): 229–246. doi:10.1007/BF02100604.
• Keel, Séan (1993). "Log abundance theorem for threefolds". Duke Math. J. 75 (1): 99–119. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Kollár, Janos (1998). Birational geometry of algebraic varieties. Cambridge University Press. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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Iitaka conjecture

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In algebraic geometry, the Iitaka conjecture is an important conjecture of birational geometry which attempts to describe the relationship between the Kodaira dimensions of the base, fibre, and total space in an algebraic fibre space.

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 κ(X) ≥ κ(Z)+κ(F), where κ denotes the Kodaira dimension.

History and status

• Iitaka

• Kawamata

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Gorenstein and Cohen–Macaulay schemes

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