# zbMATH — the first resource for mathematics

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

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J30 $$3$$-folds 14J40 $$n$$-folds ($$n>4$$)
Full Text: