Classification of higher-dimensional varieties.

*(English)*Zbl 0656.14022
Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 269-331 (1987).

[For the entire collection see Zbl 0626.00011.]

In the recent years, the classical problem to classify projective varieties up to birational equivalence has been developed a great deal by many people. This paper is an excellent and extensive survey of recent results on classification of higher-dimensional varieties. The author’s aim is mainly to review results in the classification theory (especially around the conjectures \(C_{n,m}, C^+_{n,m}\) and around varieties of Kodaira dimension 0) which do not require conjectural existence of good minimal models, and which can be considered as the natural introduction to the theory of extremal rays introduced by the author. So the paper is mostly devoted to results, not encompassed by the author’s program. The viewpoint of the theory of extremal rays is covered well by other recent survey papers by Wilson, KollĂˇr, and Kawamata, Matsuda and Matsuki.

In the recent years, the classical problem to classify projective varieties up to birational equivalence has been developed a great deal by many people. This paper is an excellent and extensive survey of recent results on classification of higher-dimensional varieties. The author’s aim is mainly to review results in the classification theory (especially around the conjectures \(C_{n,m}, C^+_{n,m}\) and around varieties of Kodaira dimension 0) which do not require conjectural existence of good minimal models, and which can be considered as the natural introduction to the theory of extremal rays introduced by the author. So the paper is mostly devoted to results, not encompassed by the author’s program. The viewpoint of the theory of extremal rays is covered well by other recent survey papers by Wilson, KollĂˇr, and Kawamata, Matsuda and Matsuki.

Reviewer: M.Beltrametti

##### MSC:

14J10 | Families, moduli, classification: algebraic theory |

14J40 | \(n\)-folds (\(n>4\)) |

14E05 | Rational and birational maps |

14J30 | \(3\)-folds |