The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L 2 -estimates of solutions of the ∂-equation by using the de Rham-Weil isomorphism between the ∂cohomology and the Čech cohomology.

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

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We give an analytic version of the injectivity theorem by using multiplier ideal sheaves, and prove some extension theorems for the adjoint bundle of dlt pairs. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry and the Strominger-Yau-Zaslow conjecture for hyperKähler manifolds.

In this paper, we study the relative anti-canonical divisor −K X/Y of an algebraic fiber space φ : X → Y , and we reveal relations among positivity conditions of −K X/Y , certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana-Cao-Matsumura's equality on Hacon-M c Kernan's question, whose original proof depends on analytics methods. The third result partially solves the question which asks whether algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.