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Torsion freeness of higher direct images of canonical bundles. (English) Zbl 0589.14016
Let X be a complex manifold, Y a reduced and irreducible complex space and let \(f:\quad X\to Y\) be a projective morphism. Assume that every connected component of X is mapped surjectively to Y. Then, the main purpose of this paper is to prove that \(R^ if_*\omega_ X\) is torsion free for every \(i\geq 0\), which is a local-analytic version of Kollár’s theorem, where \(\omega_ X\) is the canonical bundle of X. We assume furthermore that Y is non-singular, X is equidimensional and that there is an effective divisor D on Y with only normal crossings such that f is smooth over \(Y\setminus D\). Set \({\mathcal H}^ j:=R^ jf_*({\mathbb{Z}})|_{Y\setminus D}\otimes {\mathcal O}_{Y\setminus D}\) and let \(F^ p({\mathcal H}^ j)\) be the p-th Hodge filtration of \({\mathcal H}^ j\). We set \(d=\dim X-\dim Y\). Then we can show that there are isomorphisms \(\phi_ i:\quad R^ if_*\omega_{X/Y}\simeq F^ d(^ u{\mathcal H}^{d+i})\) and \(\psi_ i:\quad R^ if_*{\mathcal O}_ X\simeq Gr^ 0_ F(^{\ell}{\mathcal H}^ i)\) for every \(i\geq 0\), where \(\omega_{X/Y}=\omega_ X\otimes f^*(\omega_ Y^{-1})\) and \({}^ u{\mathcal H}^ j\) (resp. \(^{\ell}{\mathcal H}^ j)\) is the upper (resp. lower) canonical extension of the variation of Hodge structure \({\mathcal H}^ j\). In particular, \(R^ if_*\omega_ X\) and \(R^ if_*{\mathcal O}_ X\) are locally free in this case. Moreover, we have that \(R^ if_*\omega_{X/Y}\) is semi-positive and \(R^ if_*{\mathcal O}_ X\) is semi-negative for every \(i\geq 0\) if the local monodromies are unipotent.

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L20 Vanishing theorems
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32L05 Holomorphic bundles and generalizations
14C20 Divisors, linear systems, invertible sheaves
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[1] Deligne, P.: Equation différentielles à point singuliers réguliers. Lect. Notes Math. 163. Berlin, Heidelberg, New York: Springer 1970
[2] Kawamata, Y.: Kodaira dimension of algebraic fiber spaces over curves. Invent. Math.66, 57-71 (1982) · Zbl 0477.14011
[3] Kawamata, Y.: Kodaira dimension of certain algebraic fiber spaces. J. Fac. Sci. Univ. Tokyo30, 1-24 (1983) · Zbl 0516.14026
[4] Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math.123, 11-42 (1986) · Zbl 0598.14015
[5] Kollár, J.: Higher direct images of dualizing sheaves. II. Ann. Math.124, 171-202 (1986) · Zbl 0605.14014
[6] Moishezon, M.G.: Onn-dimensional compact varieties withn algebraically independent meromorphic functions. Am. Math. Soc. Trans.63, 51-117 (1967) · Zbl 0186.26204
[7] Nakamura, I.: On classification of parallelisable manifolds and small deformations. J. Differ. Geom.10, 85-112 (1975) · Zbl 0297.32019
[8] Nakayama, N.: Hodge filtrations and the higher direct images of canonical sheaves. Invent. Math.86, 217-221 (1986) · Zbl 0592.14006
[9] Nakayama, N.: On the lower semi-continuity of the plurigenera. Preprint (1985)
[10] Ohsawa, T.: Vanishing theorems on complete Kähler manifolds. Publ. RIMS Kyoto Univ.20, 21-38 (1984) · Zbl 0568.32018
[11] Saito, M.: Modules de Hodge polarisables. Preprint (1986) · Zbl 0691.14007
[12] Schmid, W.: Variation of Hodge structure. Invent. Math.22, 211-319 (1973) · Zbl 0278.14003
[13] Takegoshi, K.: Relative vanishing theorems in analytic spaces. Duke Math. J.52, 273-279 (1985) · Zbl 0577.32030
[14] Ueno, K.: On three-dimensional compact complex manifolds with non-negative Kodaira dimension. Proc. Japan Acad.56, Ser. A, 479-483 (1980) · Zbl 0486.14008
[15] Viehweg, E.: Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces. In: Advanced Studies in Pure Math. 1, Kinokuniya, Tokyo, 329-353 (1983) · Zbl 0513.14019
[16] Zucker, S.: Degenerations of Hodge bundles, in Topics in transcendental algebraic geometry. Ann. Math. Stud.106, 121-141 (1984)
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