## Higher direct images of dualizing sheaves. I.(English)Zbl 0598.14015

Let $$\pi : X \to Y$$ be a surjective morphism from a smooth projective variety $$X$$ onto a projective variety $$Y$$ over $$\mathbb{C}$$. Then the main theorem states: (1) $$R^ i\pi_*\omega_ X$$ is torsion-free for $$i\geq 0$$ and (2) $$H^ j(Y,L\otimes R^ i\pi_*\omega_ X)=0$$ for $$j>0$$ and for any ample line bundle $$L$$ on $$Y$$. This theorem is proved by using Hodge theory and the argument of S. G. Tankeev [Math. USSR, Izv. 5(1971), 29-43 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 31-44 (1971; Zbl 0212.535)]. Especially, (2) is a generalization of Kodaira’s vanishing theorem. The author gives two applications to the classification theory of higher dimensional varieties. One is the positivity property of $$\pi_*\omega_{X/Y}$$ from which we have a generalization of the characterization of abelian varieties of Y. Kawamata [Compos. Math. 43, 253-276 (1981; Zbl 0471.14022)]. The other one is to give the best possible number k such that the pluricanonical mapping $$\phi_ k$$ is stable for any projective threefolds with $$q(X)\geq 4$$ and fixed Kodaira dimension $$\kappa$$. The main theorem is further strengthened in the part II of this paper [Ann. Math., II. Ser. 124, 171- 202 (1986)] and was also generalized from the point of view of Hodge theory by H. Esnault and E. Viehweg [Invent. Math. 86, 161- 194 (1986)] and M. Saito (to appear).
Reviewer: N.Nakayama

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14E05 Rational and birational maps 32L20 Vanishing theorems 14K10 Algebraic moduli of abelian varieties, classification 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)

### Citations:

Zbl 0248.14005; Zbl 0212.535; Zbl 0471.14022
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