Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms
type: Item
instance of: scholarly article
title: Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms
zbMATH Open document ID: 0843.32018
DOI: 10.1007/BF01460997
author: Kensho Takegoshi
published in: Mathematische Annalen
publication date: 1996-08-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165374
review text: Let \(f: X \to Y\) be a proper surjective morphism from a complex manifold \(X\) of pure dimension \(n\) to a reduced analytic space \(Y\) of pure dimension \(m\), so that every connected component of \(X\) is mapped surjectively to \(Y\). Suppose \(X\) admits a Kähler metric \(\omega_X\) and \((E,h_E)\) is a Nakano semi-positive holomorphic vector bundle on \(X\) (this means that the curvature \(\Theta_h\) as \(\text{Hom} (E,E)\) valued (1,1)-form, is a positive semi-definite quadratic form on each fibre of the vector bundle \(E \otimes TX\)). The main purpose of this article is to study the Leray spectral sequence: \[ E^{p,q}_2 = H^p(Y, R^q f_* \Omega^n_X (E)) \Rightarrow H^{p + q} (X, \Omega^n_X (E)). \] The author shows the following results. Decomposition Theorem: The Leray spectral sequence for \(f\) degenerates at \(E_2\). Especially if \((X, \omega_X)\) is a compact connected Kähler manifold, then the \(\dim_C H^r (X, \Omega^n_X (E))\) is the sum over \(p + q = r\) of \(\dim_C H^p (Y, R^q f_* \Omega^n_X (E))\), for any \(r \geq 0\). Torsion freeness Theorem: For \(q \geq 1\) the sheaf homomorphism \({\mathcal L}^q\), from \(R^0 f_* \Omega^{n - q}_X(E)\) to \(R^q f_* \Omega^n_X (E)\), induced by the \(q\)-times left exterior product by \(\omega_X\) admits a splitting sheaf homomorphism \(\delta^q\), with \({\mathcal L}^q \circ \delta^q = \text{Id}\). Especially \(R^q f_* \Omega^n_X (E)\) is torsion free for \(q \geq 0\), and vanishes if \(q > n - m\). Injectivity Theorem: Let \((F,h_F)\) be a semi-positive holomorphic line bundle on \(X\) so that \(F^{\otimes j}\) admits a non-trivial holomorphic section \(\sigma\) with \(j \geq 1\). Then the sheaf homomorphism \(R^q f_* (\sigma)\), from \(R^q f_* \Omega^n_X(F^{\otimes j + k} \otimes E)\), induced by the tensor product with \(\sigma\) is injective for any \(q \geq 0\) and \(k \geq 1\). Relative vanishing Theorem: Let \(g : Y \to Z\) be a surjective proper morphism of reduced analytic spaces. Then the Leray sp…
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 840192
zbMATH Keywords: holomorphic vector bundles
zbMATH Keywords: proper Kähler morphism
zbMATH Keywords: Leray spectral sequence
reviewed by: Jesus Muciño Raymundo
MaRDI profile type: Publication
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OpenAlex ID: W2024955498
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description: scientific article; zbMATH DE number 840192