Minimal models and the Kodaira dimension of algebraic fiber spaces. (English) Zbl 0589.14014

It is known that a classical problem for algebraic varieties of dimension \(\geq 3\) is to find their minimal models and to classify them. To this aim a good definition of minimal model seems to be the following: a minimal algebraic variety is a normal projective variety having only canonical singularities whose canonical divisor is numerically effective. Moreover such a variety is called good if its canonical divisor is semi-ample. A conjecture due to the author and Reid says that every algebraic variety X with Kodaira dimension k(X)\(\geq 0\) has a birational model which is minimal and good. - On the other hand, let \(f:\quad X\to S\) be an algebraic fibre space with generic fibre \(X_{\eta}\) and let k(S)\(\geq 0\). The ”Iitaka conjecture” says that \(k(X)\geq k(X_{\eta})+Max(k(S),Var(f))\) where Var(f) is the variation of f in the sense of birational geometry.
The author proves that the Iitaka conjecture follows from the minimal model conjecture. Moreover he studies the minimal algebraic varieties X with \(k(X)=0\).
Reviewer: L.Picco Botta


14E30 Minimal model program (Mori theory, extremal rays)
14J30 \(3\)-folds
14J40 \(n\)-folds (\(n>4\))
Full Text: DOI Crelle EuDML