Higher direct images of dualizing sheaves. II. (English) Zbl 0605.14014

Kollár, János

Ann. Math. (2) 124, 171-202 (1986).

Applying the results found in part I of this paper [ibid. 123, 11-42 (1986; Zbl 0598.14015)] the author has shown that the higher direct images \({\mathcal R}^ if_*\omega_ x\) of dualizing sheaves for a projective morphism \(f: X\to Y\) is obtained as the upper-canonical extension of the variation of Hodge structures \({\mathcal R}^ jf^ 0_*{\mathbb{C}}_{X^ 0}\), if f is smooth outside a normal crossing divisor on Y. Further he obtains a decomposition theorem: \(R^{\bullet}f_*\omega_ x\cong \sum R^ if_*\omega [-i],\quad and\) studies the relation to the perverse sheaves \(P^{\bullet}(R^ jf^ 0_*{\mathbb{C}}_{X^ 0}).\)
In the last section, he discusses some conjectures arising on geometric variations of Hodge structures, which seem to be solved by the work of Morihiko Saito [cf. ”Modules de Hodge polarisables”, preprint, Res. Inst. Math. Sci. (Kyoto)].

Reviewer: N.Nakayama

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32L20	Vanishing theorems
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)

Keywords:

higher direct images of dualizing sheaves; fibrations; perverse sheaves; geometric variations of Hodge structures

Citations:

Zbl 0598.14015