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

Kollár, János

Ann. Math. (2) 123, 11-42 (1986).

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.)

Keywords:

higher direct images of dualizing sheaves; degeneration; surjective morphism; Hodge theory; vanishing theorem; classification theory of higher dimensional varieties