Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms
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 spectral sequence \[ R^p g_* R^q f_* \Omega^n_X(E) \Rightarrow R^{p + q} (g \circ f)_* \Omega^n_X (E), \] degenerates. Local freeness Theorem: suppose \(X\) is connected and \(Y\) is non-singular. (i) If \(f\) has connected fibres, then the sheaf homomorphism \({\mathcal L}^{n - m}\), from \(\Omega^m_Y\) to \(R^{n - m} f_* \Omega^n_X\), yields an isomorphism. (ii) If \(f\) is a regular family outside a normal crossing divisor of \(Y\), then \(R^q f_* \Omega^n_X\) is locally free for any \(q \geq 0\).
scientific article; zbMATH DE number 840192
scientific article